cp's OEIS Frontend

A325998 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (x + x^n)^n.

%I A325998 #11 Jul 06 2019 09:32:29
%S A325998 1,6,6,22,21,51,28,126,45,170,156,246,91,627,120,496,588,876,190,1626,
%T A325998 231,1776,1536,1392,325,4977,798,2086,3405,5025,496,8694,561,8122,
%U A325998 6636,4086,3881,21597,780,5440,11781,26016,946,24114,1035,28001,33348,8976,1225,70302,2586,36946,30501,56127,1540,66318,46698,82056,45660,16710,1891,268242,2016,20032,79806,140106,122398,171738,2415,180835,92256,249612,2701,482532,2850,32566
%N A325998 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (x + x^n)^n.
%C A325998 More generally, the following sums are equal:
%C A325998 (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C A325998 (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
%C A325998 for any fixed integer k; here, k = 3 and q = x, p = x, r = 1.
%F A325998 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (x + x^n)^n.
%F A325998 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^(n^2) / (1 - x^(n+1))^(n+3).
%e A325998 G.f.: A(x) = 1 + 6*x + 6*x^2 + 22*x^3 + 21*x^4 + 51*x^5 + 28*x^6 + 126*x^7 + 45*x^8 + 170*x^9 + 156*x^10 + 246*x^11 + 91*x^12 + 627*x^13 + 120*x^14 +...
%e A325998 where
%e A325998 A(x) = 1 + 3*(x + x) + 6*(x + x^2)^2 + 10*(x + x^3)^3 + 15*(x + x^4)^4 + 21*(x + x^5)^5 + 28*(x + x^6)^6 + 36*(x + x^7)^7 + 45*(x + x^8)^8 + 55*(x + x^9)^9 + ...
%e A325998 Also
%e A325998 A(x) = 1/(1-x)^3 + 3*x/(1 - x^2)^4 + 6*x^4/(1 - x^3)^5 + 10*x^9/(1 - x^4)^6 + 15*x^16/(1 - x^5)^7 + 21*x^25/(1 - x^6)^8 + 28*x^36/(1 - x^7)^9 + 36*x^49/(1 - x^8)^10 + 45*x^64/(1 - x^9)^11 + 55*x^81/(1 - x^10)^12 + ...
%o A325998 (PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)/2 * (x + x^m +x*O(x^n))^m), n)}
%o A325998 for(n=0, 100, print1(a(n), ", "))
%o A325998 (PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)/2 * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+3)), n)}
%o A325998 for(n=0, 100, print1(a(n), ", "))
%Y A325998 Cf. A217669, A325997, A325999.
%K A325998 nonn
%O A325998 0,2
%A A325998 _Paul D. Hanna_, Jun 02 2019