Original entry on oeis.org
1, 2, 9, 70, 805, 12646, 257179, 6524176, 200811433, 7340612842, 313294235311, 15395868322660, 861109521894637, 54282864059246590, 3824491871326292755, 298974154411140942856, 25767887775430753766353, 2434836258338521063652050
Offset: 1
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{a(n)=local(Egf); Egf=sum(m=0, n, x^m*exp(m*(m+1)/2*x+x*O(x^n))); if(n<1,0,(n-1)!*polcoeff(Egf, n))}
A193421
E.g.f.: Sum_{n>=0} x^n * exp(n^2*x).
Original entry on oeis.org
1, 1, 4, 33, 436, 8185, 206046, 6622945, 263313688, 12627149265, 716160702970, 47284266221401, 3587061106583604, 309251317536586633, 30017652739792964806, 3254137305364883664945, 391238883136463492841136, 51846176797206158144925985
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 436*x^4/4! + 8185*x^5/5! + 206046*x^6/6! +...
where
A(x) = 1 + x*exp(x) + x^2*exp(4*x) + x^3*exp(9*x) + x^4*exp(16*x) +...
By a q-series identity:
A(x) = 1 + x*exp(x)*(1-x*exp(x))/(1-x*exp(3*x)) + x^2*exp(2*x)*(1-x*exp(x))*(1-x*exp(5*x))/((1-x*exp(3*x))*(1-x*exp(7*x))) + x^3*exp(3*x)*(1-x*exp(x))*(1-x*exp(5*x))*(1-x*exp(9*x))/((1-x*exp(3*x))*(1-x*exp(7*x))*(1-x*exp(11*x))) +...
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Flatten[{1,Table[n! * Sum[(n-k)^(2*k)/k!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 21 2014 *)
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{a(n)=local(Egf); Egf=sum(m=0, n, x^m*exp(m^2*x+x*O(x^n))); n!*polcoeff(Egf, n)}
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/* q-series identity: */
{a(n)=local(A=1+x);for(i=1, n, A=sum(m=0, n, x^m*exp(m*x+x*O(x^n))*prod(k=1, m, (1-x*exp((4*k-3)*x+x*O(x^n)))/(1-x*exp((4*k-1)*x+x*O(x^n)))))); n!*polcoeff(A, n)}
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{a(n) = n!*sum(k=0,n, (n-k)^(2*k)/k!)}
for(n=0,20,print1(a(n),", "))
A193466
E.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} cosh(k*x).
Original entry on oeis.org
1, 1, 2, 9, 84, 965, 12750, 225967, 5241880, 139776345, 4272148890, 155402034491, 6513558987540, 304210965928597, 15965624278036342, 941149313037711975, 61160783460181817520, 4356686998946564113457, 340627068039399668576946, 29015657457166019702796787
Offset: 0
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 84*x^4/4! + 965*x^5/5! + 12750*x^6/6! + 225967*x^7/7! +...
where
A(x) = 1 + x*cosh(x) + x^2*cosh(x)*cosh(2*x) + x^3*cosh(x)*cosh(2*x)*cosh(3*x) + x^4*cosh(x)*cosh(2*x)*cosh(3*x)*cosh(4*x) +...
Also,
A(x) = 1 + x*exp(-x)*(1+exp(2*x))/2 + x^2*exp(-3*x)*(1+exp(2*x))*(1+exp(4*x))/2^2 + x^3*exp(-6*x)*(1+exp(2*x))*(1+exp(4*x))*(1+exp(6*x))/2^3 +...
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{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, x^m*prod(k=1, m, cosh(k*X))); n!*polcoeff(Egf, n)}
Original entry on oeis.org
1, 1, 3, 21, 193, 2125, 32281, 655235, 15530705, 427214889, 14127457681, 542796582295, 23400843532969, 1140401734145453, 62743287535847465, 3822548966261363595, 256275705820386124321, 18923726002188870476497
Offset: 1
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{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, x^m*prod(k=1, m, cosh(k*X))); if(n<1,0,(n-1)!*polcoeff(Egf, n))}
Showing 1-4 of 4 results.
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