with(combinat);with(LinearAlgebra):with(plots):
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rown:=proc(n); local a,j; a:=Vector(n+1); for j from 0 to n do a[j+1]:=numbcomb(n,j): od; RETURN(Transpose(a)); end proc;
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rown(7)
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pasn:=proc(n); local a,j,k; a:=Matrix(n+1); for k from 0 to n do for j from 0 to n do a[j+1,k+1]:=numbcomb(j,k): od; od; RETURN(a); end proc;
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pasn(5)
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MatrixInverse(pasn(5));
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numbcomb(3,5);
IiIh
prodn:=proc(n); local a,j; a:=1; for j from 0 to n do a:=a*numbcomb(n,j): od; RETURN(a); end proc;
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prodn(0);
IiIi
prodn(4);
IiMnKg==
prodn(5);
IiUrRA==
seq(prodn(j),j=1..8);
NioiIiIiIiMiIioiIycqIiUrRCInKz87IilENVpFIiw/Ymo5NSI=
pl:=[]:for j from 1 to 20 do pl:=[op(pl),[j,evalf(prodn(j+1)*prodn(j-1)/(prodn(j)^2))]]; od: pl;
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pointplot(pl);
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for j from 1 to 10 do [expand( (1+1/j)^j), prodn(j+1)*prodn(j-1)/(prodn(j)^2)]; od;
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limit((1+1/x)^x,x=infinity);
LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiI=
plot((1+1/x)^x,x=1..20);
6*-%'CURVESG6$7dw7$$"#5!""$"#?!""7$$"2oQxa\%*)*4"!#;$"18-&[$*zl.#!#:7$$"2kHf=;7o="!#;$"2tO0PD^a1#!#;7$$"2NqS"o,c%G"!#;$"2K]$*G='3&4#!#;7$$"2;He;6cHQ"!#;$"289#eg?GA@!#;7$$"22;KkR%)3["!#;$"20?N#=<1Z@!#;7$$"2Eb5@Xw;d"!#;$"29!Q^&GV#o@!#;7$$"1AW)oO'ol;!#:$"2:^NzGl&)=#!#;7$$"228E_b6Hw"!#;$"2c#zE>q/3A!#;7$$"2)yd:T\#)f=!#;$"1(*3$*H#zgA#!#:7$$"2*oPvqA^f>!#;$"2(**zc#yMLC#!#;7$$"1BY#\#pJZ?!#:$"22f#eAYbdA!#;7$$"2kAX!pT;Y@!#;$"2vEG:#4dsA!#;7$$"1OsW*G<aC#!#:$"2cc*oV'*o'G#!#;7$$"1`06ig1TB!#:$"1fA,X*p%*H#!#:7$$"2`-05JCzU#!#;$"0<%y$)QV5B!#97$$"29D]+<27`#!#;$"2%*yN/6VFK#!#;7$$"2C['H*H,(=E!#;$"1L2<O8gKB!#:7$$"2x`2:xx/s#!#;$"0ejA\lMM#!#97$$"1pQx%>n0"G!#:$"28:(y`>e_B!#;7$$"18D]q&4%4H!#:$"1NLHRa3iB!#:7$$"2(4>Q')3`.I!#;$"2F,1UY&oqB!#;7$$"1Qw_0mt,J!#:$"29Ee(*[D#zB!#;7$$"1b5@7,#>>$!#:$"2EsFoM0nQ#!#;7$$"2#)oPv?&>*G$!#;$"2`iC<)4T%R#!#;7$$"2Ed9HpO-R$!#;$"1.zn%pV?S#!#:7$$"1C['Hg$>yM!#:$"1Nni\-S3C!#:7$$"1(QxaZ*=tN!#:$"2x#y:%)Q)\T#!#;7$$"1yc82&H8n$!#:$"2FH;)ys\@C!#;7$$"0/3;QStw$!#9$"2XFahj.wU#!#;7$$"1OsWReBgQ!#:$"2w_yt%oFLC!#;7$$"2%yc82/QjR!#;$"1kl;I:KRC!#:7$$"2&=Pu)Qgg0%!#;$"2%zB*G#z`WC!#;7$$"1"Gc70;]:%!#:$"1w"\h,'*)\C!#:7$$"2&zf>*e%oWU!#;$"2YiZC%HdaC!#;7$$"1W'Gd)frUV!#:$"2maLVs-&fC!#;7$$"1-05!*e&\V%!#:$"2.bAVgvRY#!#;7$$"1Z%*)y;t8`%!#:$"2Ysn*[%)[oC!#;7$$"0,-/oQci%!#9$"2'pNX["[FZ#!#;7$$"1['Hf$fKCZ!#:$"2(**Gk3X0xC!#;7$$"118E#\t$>[!#:$"2dD')=Uh5[#!#;7$$"1mJjYLd;\!#:$"2B>;/_B][#!#;7$$"1yc8P$oH,&!#:$"2#o/wkS#))[#!#;7$$"1b5@A`a,^!#:$"1*y(3h'3A\#!#:7$$"1v],VX1._!#:$"21]&z!*o'f\#!#;7$$"2vg@VakQH&!#;$"1fl5oQA*\#!#:7$$"1_.2%Gw1R&!#:$"2&=iJpGf-D!#;7$$"1[&4>ZPL[&!#:$"1o%G1o@d]#!#:7$$"1<Lm7eA'e&!#:$"1cw:Y544D!#:7$$"1V'Gdb>`n&!#:$"1;0QcM#>^#!#:7$$"1D]+^<Wwd!#:$"1@$QClY]^#!#:7$$"1b3<aOgoe!#:$"1e]"o!="y^#!#:7$$"017CH9%pf!#9$"2:124Z^2_#!#;7$$"106ACoRcg!#:$"1>#>JU>K_#!#:7$$"1nLnu(fa:'!#:$"0)>_6g&f_#!#97$$"1d8F%p\6D'!#:$"1Gs<Gu_GD!#:7$$"1*zf>5xnM'!#:$"2nY6JKH5`#!#;7$$"1U$oOr_?W'!#:$"2NdDI?dM`#!#;7$$"2X)pR\FeLl!#;$"1m")\D5tND!#:7$$"005?9HDj'!#9$"1AVNOp7QD!#:7$$"1oMpGA$os'!#:$"14eShDNSD!#:7$$"21<Moy"3Eo!#;$"1K'p`DOEa#!#:7$$"1AW)o`Ef"p!#:$"2Ye4Ba`Ya#!#;7$$"1@T#[!=A:q!#:$"20L.yvHoa#!#;7$$"1nMpo$H.6(!#:$"1DuF<O')[D!#:7$$"1_/4Qa?0s!#:$"1-h1")\%3b#!#:7$$"13:IgQI/t!#:$"29N!=1c'Gb#!#;7$$"1sV([7xbR(!#:$"10L7zRoaD!#:7$$"1>QwKM0*[(!#:$"1wYJU_]cD!#:7$$"1$f=PQiAf(!#:$"1="G^dp%eD!#:7$$"1u[(\j4do(!#:$"2;Q.RR2-c#!#;7$$"1w],`pF"y(!#:$"2;&egdf%>c#!#;7$$"1'4>QXm%yy!#:$"2$=,PB_njD!#;7$$"1*oPv^-y'z!#:$"2%\)>QLJ_c#!#;7$$"19Fao=.j!)!#:$"2[f\Q&f&oc#!#;7$$"1)oPvKOv:)!#:$"1:UD[WVoD!#:7$$"0e:JsE'e#)!#9$"2W@&y,p3qD!#;7$$"0&**)zhMyM)!#9$"1r3"p7::d#!#:7$$"1\(\*4?w]%)!#:$"1>\ed"HJd#!#:7$$"1Y"He/YOa)!#:$"1hav3bbuD!#:7$$"1d8FW*\bj)!#:$"2C5Q@ZRfd#!#;7$$"1'[(\fhPM()!#:$"2yQ,(H#)RxD!#;7$$"1zc8Z@gL))!#:$"12#Q45L)yD!#:7$$"1W)oP5$)Q#*)!#:$"1$yD(\M6!e#!#:7$$"1PsWpuv>!*!#:$"2/c4.iZ9e#!#;7$$"*[@L6*!")$"1[B`?]s#e#!#:7$$"18E_aeu9#*!#:$"2)RXibI3%e#!#;7$$"1Qu[Ph]-$*!#:$"1<ybbjB&e#!#:7$$"1v],8?I.%*!#:$"2$=0GMn`'e#!#;7$$"1C^-Na0)\*!#:$"2Wtf;=Oxe#!#;7$$"1d:JUO*=f*!#:$"2%ozY.F!*)e#!#;7$$"1kHf)\1eo*!#:$"2.Jw&e%\+f#!#;7$$"1MpQn;f"y*!#:$"2j.XgJ)>"f#!#;7$$"1"He;MAJ))*!#:$"2c!Go9QR#f#!#;7$$"1%pQx-;i(**!#:$"2$ywvl.Z$f#!#;7$$"2#\)p\96o+"!#:$"2PEul9:Xf#!#;7$$"2MoOVhMm,"!#:$"0[TC#Gh&f#!#97$$"2nMp))pOk-"!#:$"2w;ng'))o'f#!#;7$$"2e9H=lM_."!#:$"2W9VNsQwf#!#;7$$"1*yd:Ycc/"!#9$"1*[@p'[u)f#!#:7$$"2iE`'[pPa5!#:$"2s-8K'\l*f#!#;7$$"2`4>y`wX1"!#:$"2Lbo**)>q+E!#;7$$"2(Rzed5Ju5!#:$"1#zdp<%o,E!#:7$$"228EU#G+$3"!#:$"2xP:]QZDg#!#;7$$"28Fa[ixF4"!#:$"1#[+`+.Ng#!#:7$$"0.1#>sh-6!#8$"22d4(["\Wg#!#;7$$"1<MoZ+T76!#9$"1^TpHaP0E!#:7$$"2jD^KD*[@6!#:$"1Fw3T3A1E!#:7$$"2KkGZC!*38"!#:$"1$\tL(H32E!#:7$$"19GcjFhS6!#9$"2MF1Yugzg#!#;7$$"2)yd:-TI]6!#:$"1&\vQ,A)3E!#:7$$"2zd:^$GFg6!#:$"2VDuG,%p4E!#;7$$"2KmK0I`!p6!#:$"1^U^R1X5E!#:7$$"2OsW\-Q*y6!#:$"2o+N/!**G6E!#;7$$"2Z#\)pLj))="!#:$"2WF.;c>@h#!#;7$$"2kD^U@G%)>"!#:$"1&Qum+2Hh#!#:7$$"2NqS"RS627!#:$"0bKr-7Oh#!#97$$"1E_/DBW<7!#9$"0N6-=QWh#!#97$$"2#\)pzt">E7!#:$"1kDF-z7:E!#:7$$"2\&4>&QpjB"!#:$"1(f67x=fh#!#:7$$"2ye<vKy`C"!#:$"1LS8s(3mh#!#:7$$"2AX!4lDEb7!#:$"1)36q=bth#!#:7$$"2>RympuYE"!#:$"1jrTKe0=E!#:7$$"2Z'Hfo_\u7!#:$"2'R/`%ew(=E!#;7$$"2lIh#>O^$G"!#:$"2Y?!Q(QH%>E!#;7$$"2*pRzG6C$H"!#:$"2U#oz')R7?E!#;7$$"2$e;Lx_M.8!#:$"2;)*4w@N3i#!#;7$$"2NoO$o4978!#:$"22Kye+Y9i#!#;7$$"2'Rzeb0k@8!#:$"2#Q-tHr4AE!#;7$$"2(oPveXXJ8!#:$".f-ngFi#!#77$$"2^+,ikb5M"!#:$"27q>P,,Mi#!#;7$$"2Y#\)>>X.N"!#:$"1CO9eC,CE!#:7$$"2)oPv['f1O"!#:$"22#e'R>#oCE!#;7$$"2Gd9pkF*p8!#:$"2&=zS9fFDE!#;7$$"1He;8K#)z8!#9$"1mIqx:!fi#!#:7$$"1*zfp1!z)Q"!#9$"1F/b%Hhki#!#:7$$"2^18m?$f)R"!#:$"2n)=%=]lqi#!#;7$$"28D]q><yS"!#:$"2%RHJ'zEwi#!#;7$$"2e9H[#*euT"!#:$"2D^%fsh?GE!#;7$$"2=S!3wa)oU"!#:$"2x!pc-bwGE!#;7$$"1mJj,UvO9!#9$"2)enHtOMHE!#;7$$"2:Lms&*eiW"!#:$"1;&HC]$*)HE!#:7$$"2u^.F%*yfX"!#:$"2ku!G%z[/j#!#;7$$"2)oPvT%=cY"!#:$"20Q/*QE*4j#!#;7$$"2jIh-9wWZ"!#:$"1lYsqk[JE!#:7$$"2$3<Mi!GY["!#:$"1LI.Hc/KE!#:7$$"2>OsC13P\"!#:$"2B'QNx'RDj#!#;7$$"1OsWO#*Q.:!#9$"0-AuBgIj#!#97$$"2e:J_NbE^"!#:$"2CBlHf_Nj#!#;7$$"2E`1$*=WH_"!#:$"2Cf0lm#4ME!#;7$$"2^18Oc`=`"!#:$"1*HD!)zaXj#!#:7$$"2NqSJyl>a"!#:$"2i>v9@t]j#!#;7$$"2iGdM(>=^:!#:$"2D.%)z7Sbj#!#;7$$"2pS"GPIEh:!#:$"1ysC'*[/OE!#:7$$"2;Ji/Hh*p:!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stir1m:=proc(n); local a,j,k; a:=Matrix(n+1); for k from 0 to n do for j from 0 to n do a[j+1,k+1]:=stirling1(j,k): od; od; RETURN(a); end proc;
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdzdGlyMW1HRihmKjYjSSJuR0YoNiVJImFHRihJImpHRihJImtHRihGKEYoQyU+OCQtSSdNYXRyaXhHRiU2IywmOSQiIiJGPUY9Pyg4JiIiIUY9RjxJJXRydWVHRiY/KDglRkBGPUY8RkE+JkY3NiQsJkZDRj1GPUY9LCZGP0Y9Rj1GPS1fSSljb21iaW5hdEdGJUkqc3RpcmxpbmcxR0YoNiRGQ0Y/LUknUkVUVVJOR0YmNiNGN0YoRihGKDcjRi4=
stir2m:=proc(n); local a,j,k; a:=Matrix(n+1); for k from 0 to n do for j from 0 to n do a[j+1,k+1]:=stirling2(j,k): od; od; RETURN(a); end proc;
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdzdGlyMm1HRihmKjYjSSJuR0YoNiVJImFHRihJImpHRihJImtHRihGKEYoQyU+OCQtSSdNYXRyaXhHRiU2IywmOSQiIiJGPUY9Pyg4JiIiIUY9RjxJJXRydWVHRiY/KDglRkBGPUY8RkE+JkY3NiQsJkZDRj1GPUY9LCZGP0Y9Rj1GPS1fSSljb21iaW5hdEdGJUkqc3RpcmxpbmcyR0YoNiRGQ0Y/LUknUkVUVVJOR0YmNiNGN0YoRihGKDcjRi4=
#stir1m(7);
stir2m(7);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1JykpZS9PdVN1WSU9LUknTUFUUklYR0YoNiM3KjcqLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIxRigtRjY2I1EiMEYoRjxGPEY8RjxGPEY8NypGPEY1RjxGPEY8RjxGPEY8NypGPEY1RjVGPEY8RjxGPEY8NypGPEY1LUY2NiNRIjNGKEY1RjxGPEY8Rjw3KkY8RjUtRjY2I1EiN0YoLUY2NiNRIjZGKEY1RjxGPEY8NypGPEY1LUY2NiNRIzE1RigtRjY2I1EjMjVGKC1GNjYjUSMxMEYoRjVGPEY8NypGPEY1LUY2NiNRIzMxRigtRjY2I1EjOTBGKC1GNjYjUSM2NUYoRk1GNUY8NypGPEY1LUY2NiNRIzYzRigtRjY2I1EkMzAxRigtRjY2I1EkMzUwRigtRjY2I1EkMTQwRigtRjY2I1EjMjFGKEY1SSdNYXRyaXhHRiU3Iy1Gam82Iy9JJCVpZEdGKEYv
#MatrixInverse(stir2(7));
stirling1(4,1);
ISIn
expand(x+3*x*(x-1)+x*(x-1)*(x-2));
KiRJInhHNiIiIiQ=
expand(x+7*x*(x-1)+6*x*(x-1)*(x-2)+1*x*(x-1)*(x-2)*(x-3));
KiRJInhHNiIiIiU=
s
expand(x*(x-1)*(x-2)*(x-3)*(x-4));
LCwqJEkieEc2IiIiJiIiIiokRiQiIiUhIzUqJEYkIiIkIiNOKiRGJCIiIyEjXUYkIiND
expand(x*(x+1)*(x+2)*(x+3)*(x+4));
LCwqJEkieEc2IiIiJiIiIiokRiQiIiUiIzUqJEYkIiIkIiNOKiRGJCIiIyIjXUYkIiND