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A326004 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^n * (1 + x^n)^n.

%I A326004 #8 Jul 02 2019 22:38:59
%S A326004 1,4,14,20,55,56,154,120,305,280,566,364,1189,560,1520,1376,2429,1140,
%T A326004 4570,1540,5226,4544,6304,2600,14685,3556,10934,11980,18215,4960,
%U A326004 31882,5984,31289,27160,27150,12636,82093,9880,39920,55160,93631,13244,121178,15180,126875,130696,78224,19600,316645,22940,165386,179844,281399,27720,370090,150976,410629,297560,179830,37820,1208458,41664,229184,489280,801305,450516,987482,54740
%N A326004 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^n * (1 + x^n)^n.
%C A326004 More generally, the following sums are equal:
%C A326004 (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C A326004 (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
%C A326004 for any fixed integer k; here, k = 4 and p = 1, q = x, r = x.
%F A326004 Generating functions.
%F A326004 (1) Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^n * (1 + x^n)^n.
%F A326004 (2) Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^(n*(n+1)) / (1 - x^(n+1))^(n+4).
%e A326004 G.f.: A(x) = 1 + 4*x + 14*x^2 + 20*x^3 + 55*x^4 + 56*x^5 + 154*x^6 + 120*x^7 + 305*x^8 + 280*x^9 + 566*x^10 + 364*x^11 + 1189*x^12 + 560*x^13 + 1520*x^14 + 1376*x^15 + 2429*x^16 + 1140*x^17 + 4570*x^18 + 1540*x^19 + 5226*x^20 + ...
%e A326004 where we have the following series identity:
%e A326004 A(x) = 1 + 4*x*(1+x) + 10*x^2*(1+x^2)^2 + 20*x^3*(1+x^3)^3 + 35*x^4*(1+x^4)^4 + 56*x^5*(1+x^5)^5  + 84*x^6*(1+x^6)^6 + 120*x^7*(1+x^7)^7 + 165*x^8*(1+x^8)^8 + 220*x^9*(1+x^9)^9 +...
%e A326004 is equal to
%e A326004 A(x) = 1/(1-x)^4 + 4*x^2/(1-x^2)^5 + 10*x^6/(1-x^3)^6 + 20*x^12/(1-x^4)^7 + 35*x^20/(1-x^5)^8 + 56*x^30/(1-x^6)^9 + 84*x^42/(1-x^7)^10 + 120*x^56/(1-x^8)^11 +...
%o A326004 (PARI) {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)*(m+3)/3! * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
%o A326004 for(n=0,120,print1(a(n),", "))
%o A326004 (PARI) {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)*(m+3)/3! * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+4))); polcoeff(A,n)}
%o A326004 for(n=0,120,print1(a(n),", "))
%Y A326004 Cf. A217668 (k=1), A326002 (k=2), A326003 (k=3), A326005 (k=5).
%K A326004 nonn
%O A326004 0,2
%A A326004 _Paul D. Hanna_, Jun 01 2019