|
%I #16 Jun 09 2023 09:14:37
%S 1,1,2,4,10,28,88,306,1158,4730,20722,96776,479340,2507510,13804014,
%T 79718782,481614806,3036358968,19932689952,135981543762,962319171782,
%U 7053068549250,53458038451082,418440466421960,3378290373259300,28099682071640734,240537280709926718
%N G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + (n-k+1)*x) / (1 + k*x + (n-k+1)*x^2).
%C Compare to the following identities, which hold for any fixed b and c:
%C (1) Sum_{n>=0} x^n * Product_{k=1..n} (b + k*x)/(1 + b*x + k*x^2) = (1 + b*x)/(1 - x^2).
%C (2) Sum_{n>=0} x^n * Product_{k=1..n} (k + c*x)/(1 + k*x + c*x^2) = (1 + c*x^2)/(1 - x).
%C (3) Sum_{n>=0} x^n * Product_{k=1..n} (b*k + c*k*x)/(1 + b*k*x + c*k*x^2) = 1/(1 - b*x - c*x^2).
%C Conjectures:
%C (1) a(6*n + k) == 0 (mod 4) for n > 0 when k = {0,5},
%C (2) a(6*n + k) == 2 (mod 4) for n > 0 when k = {1,2,3,4}.
%H Paul D. Hanna, <a href="/A363110/b363110.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be described by the following.
%F (1) Sum_{n>=0} x^n * Product_{k=1..n} (k + (n-k+1)*x) / (1 + k*x + (n-k+1)*x^2).
%F (2) Sum_{n>=0} x^n * (Sum_{k=0..n} A067948(n,k) * x^k) / Product_{k=1..n} (1 + k*x + (n-k+1)*x^2).
%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 88*x^6 + 306*x^7 + 1158*x^8 + 4730*x^9 + 20722*x^10 + 96776*x^11 + 479340*x^12 + ...
%e where
%e A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1 + 2*x)*(2 + x)/((1 + x + 2*x^2)*(1 + 2*x + x^2)) + x^3*(1 + 3*x)*(2 + 2*x)*(3 + x)/((1 + x + 3*x^2)*(1 + 2*x + 2*x^2)*(1 + 3*x + x^2)) + x^4*(1 + 4*x)*(2 + 3*x)*(3 + 2*x)*(4 + x)/((1 + x + 4*x^2)*(1 + 2*x + 3*x^2)*(1 + 3*x + 2*x^2)*(1 + 4*x + x^2)) + ...
%o (PARI) {a(n) = polcoeff( A = sum(m=0, n, x^m*prod(k=1, m, (k + (m-k+1)*x)/(1 + k*x + (m-k+1)*x^2 +x*O(x^n))) ), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A204064, A204066, A316370, A067948.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 02 2023
| 