A118804
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1+k*x)^2.
Original entry on oeis.org
1, 2, 9, 66, 685, 9294, 156697, 3169910, 74998081, 2035262154, 62391632417, 2134187066010, 80641239109677, 3337651407273846, 150239268816661137, 7310140430519234862, 382439924662714479457, 21413128578896024921298, 1277905479699750127195097
Offset: 0
1 = 1/(1+x)^2 + 2*x/((1+x)*(1+2*x))^2 + 9*x^2/((1+x)*(1+2*x)*(1+3*x))^2 + 66*x^3/((1+x)*(1+2*x)*(1+3*x)*(1+4*x))^2 +...+ a(n)*x^n/((1+x)*(1+2x)*(1+3x)*...*(1+n*x))^2 +...
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{a(n)=if(n==0, 1, polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, 1+j*x+x*O(x^n))^2), n))}
A187806
G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n * Product_{k=1..n+1} (1-k*x).
Original entry on oeis.org
1, 2, 7, 39, 314, 3388, 46409, 776267, 15406059, 354928082, 9330754204, 276092552520, 9092298247070, 330151121828252, 13114259187006717, 566025800996830823, 26391137839213285415, 1322515573450223865750, 70912312814053387968103, 4052279260763983306587339
Offset: 0
1/(1-x) = (1-x) + 2*x*(1-x)*(1-2*x) + 7*x^2*(1-x)*(1-2*x)*(1-3*x) + 39*x^3*(1-x)*(1-2*x)*(1-3*x)*(1-4*x) + 314*x^4*(1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x) + 3388*x^5*(1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x) +...
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{a(n)=if(n==0, 1, 1-polcoeff(sum(k=0, n-1, a(k)*x^k*prod(j=1, k+1, 1-j*x+x*O(x^n))), n))}
for(n=0,20,print1(a(n),", "))
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upto(n) = my(v1); v1 = vector(n+1, i, 1); for(i=1, n, for(j=i+1, n+1, v1[j] += i*v1[j-1])); v1 \\ Mikhail Kurkov, Oct 25 2024
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