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A326003 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1 + x^n)^n.

%I A326003 #11 Nov 28 2023 09:27:22
%S A326003 1,3,9,10,27,21,64,36,105,85,171,78,359,105,372,346,573,171,1105,210,
%T A326003 1116,1009,1134,300,3237,456,1743,2386,3375,465,5947,528,5529,4885,
%U A326003 3537,1926,14917,741,4770,9010,16551,903,17963,990,19977,22291,8028,1176,49925,1527,23961,24634,41289,1485,48502,27336,58809,37621,15255,1830,184218,1953,18384,59830,106137,77286,121705,2346,140115,78385,159846,2628,346846,2775,30267,293866
%N A326003 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1 + x^n)^n.
%C A326003 More generally, the following sums are equal:
%C A326003 (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C A326003 (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
%C A326003 for any fixed integer k; here, k = 3 and p = 1, q = x, r = x.
%H A326003 Paul D. Hanna, <a href="/A326003/b326003.txt">Table of n, a(n) for n = 0..2050</a>
%F A326003 Generating functions.
%F A326003 (1) Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1 + x^n)^n.
%F A326003 (2) Sum_{n>=0} (n+1)*(n+2)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+3).
%e A326003 G.f.: A(x) = 1 + 3*x + 9*x^2 + 10*x^3 + 27*x^4 + 21*x^5 + 64*x^6 + 36*x^7 + 105*x^8 + 85*x^9 + 171*x^10 + 78*x^11 + 359*x^12 + 105*x^13 + 372*x^14 + 346*x^15 + 573*x^16 + 171*x^17 + 1105*x^18 + 210*x^19 + 1116*x^20 + ...
%e A326003 where we have the following series identity:
%e A326003 A(x) = 1 + 3*x*(1+x) + 6*x^2*(1+x^2)^2 + 10*x^3*(1+x^3)^3 + 15*x^4*(1+x^4)^4 + 21*x^5*(1+x^5)^5  + 28*x^6*(1+x^6)^6 + 36*x^7*(1+x^7)^7 + 45*x^8*(1+x^8)^8 + 55*x^9*(1+x^9)^9 +...
%e A326003 is equal to
%e A326003 A(x) = 1/(1-x)^3 + 3*x^2/(1-x^2)^4 + 6*x^6/(1-x^3)^5 + 10*x^12/(1-x^4)^6 + 15*x^20/(1-x^5)^7 + 21*x^30/(1-x^6)^8 + 28*x^42/(1-x^7)^9 + 36*x^56/(1-x^8)^10 +...
%o A326003 (PARI) {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)/2 * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
%o A326003 for(n=0,120,print1(a(n),", "))
%o A326003 (PARI) {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)/2 * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+3))); polcoeff(A,n)}
%o A326003 for(n=0,120,print1(a(n),", "))
%Y A326003 Cf. A217668 (k=1), A326002 (k=2), A326004 (k=4), A326005 (k=5).
%K A326003 nonn
%O A326003 0,2
%A A326003 _Paul D. Hanna_, Jun 01 2019