A187806 - OEIS

%I #16 Oct 25 2024 06:33:41

%S 1,2,7,39,314,3388,46409,776267,15406059,354928082,9330754204,

%T 276092552520,9092298247070,330151121828252,13114259187006717,

%U 566025800996830823,26391137839213285415,1322515573450223865750,70912312814053387968103,4052279260763983306587339

%N G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n * Product_{k=1..n+1} (1-k*x).

%C Compare to: 1 = Sum_{n>=0} A082161(n)*x^n * Product_{k=1..n+1} (1-k*x).

%H Fedor Petrov, <a href="https://mathoverflow.net/a/477153/231922">On a generating function and vector ν of length n</a>, answer to question on MathOverflow (2024).

%e 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) +...

%o (PARI) {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))}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) 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

%Y Cf. A082161, A118805.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 06 2013