cp's OEIS Frontend

A325999 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * (x + x^n)^n.

%I A325999 #15 Jul 06 2019 09:33:59
%S A325999 1,8,10,40,45,116,84,320,165,520,496,868,455,2100,680,2136,2264,3680,
%T A325999 1330,6920,1771,7988,6920,8060,2925,22732,4914,13580,17365,26440,5456,
%U A325999 46212,6545,45000,37800,32376,20773,119660,10660,46900,74221,143528,14190,161540,16215,177196,194764,89800,20825,447040,28046,239928,229725,384860,29260,492128,257734,569140,372480,201500,39711,1763416,43680,255200,639430,1068856,733074,1337080
%N A325999 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * (x + x^n)^n.
%C A325999 More generally, the following sums are equal:
%C A325999 (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C A325999 (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
%C A325999 for any fixed integer k; here, k = 4 and q = x, p = x, r = 1.
%F A325999 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * (x + x^n)^n.
%F A325999 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^(n^2) / (1 - x^(n+1))^(n+4).
%F A325999 FORMULAS FOR TERMS.
%F A325999 a(5*n + 2) = 0 (mod 5),
%F A325999 a(5*n + 3) = 0 (mod 5),
%F A325999 a(5*n + 4) = 0 (mod 5), for n >= 0.
%e A325999 G.f.: A(x) = 1 + 8*x + 10*x^2 + 40*x^3 + 45*x^4 + 116*x^5 + 84*x^6 + 320*x^7 + 165*x^8 + 520*x^9 + 496*x^10 + 868*x^11 + 455*x^12 + 2100*x^13 + 680*x^14 +...
%e A325999 where
%e A325999 A(x) = 1 + 4*(x + x) + 10*(x + x^2)^2 + 20*(x + x^3)^3 + 35*(x + x^4)^4 + 56*(x + x^5)^5 + 84*(x + x^6)^6 + 120*(x + x^7)^7 + 165*(x + x^8)^8 + 220*(x + x^9)^9 + ...
%e A325999 Also
%e A325999 A(x) = 1/(1-x)^4 + 4*x/(1 - x^2)^5 + 10*x^4/(1 - x^3)^6 + 20*x^9/(1 - x^4)^7 + 35*x^16/(1 - x^5)^8 + 56*x^25/(1 - x^6)^9 + 84*x^36/(1 - x^7)^10 + 120*x^49/(1 - x^8)^11 + 165*x^64/(1 - x^9)^12 + 220*x^81/(1 - x^10)^13 + ...
%o A325999 (PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)*(m+3)/3! * (x + x^m +x*O(x^n))^m), n)}
%o A325999 for(n=0, 100, print1(a(n), ", "))
%o A325999 (PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)*(m+3)/3! * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+4)), n)}
%o A325999 for(n=0, 100, print1(a(n), ", "))
%Y A325999 Cf. A217669, A325997, A325998.
%K A325999 nonn
%O A325999 0,2
%A A325999 _Paul D. Hanna_, Jun 02 2019