cp's OEIS Frontend

A326012 a(n) = (n+1)*(n+2)/2 * (2^n + 1)^n.

%I A326012 #9 Jun 02 2025 16:06:21
%S A326012 1,9,150,7290,1252815,821843253,2111728937500,21400822889676324,
%T A326012 856401659982049536045,135337757314088315015913015,
%U A326012 84485579916959616279602050781250,208476038116104541762965430756506783822,2035321216424903698212658521136507048387952731,78695103743782916150566765682074995597941860823900265,12062337788631745341740759075565225314748425863190185546875000
%N A326012 a(n) = (n+1)*(n+2)/2 * (2^n + 1)^n.
%C A326012 More generally, the following sums are equal:
%C A326012 (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C A326012 (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
%C A326012 for any fixed integer k; here, k = 3 and p = 1, q = 2, r = x.
%F A326012 O.g.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (2^n + 1)^n * x^n.
%F A326012 O.g.f.: Sum_{n>=0} (n+1)*(n+2)/2 * 2^(n^2) * x^n / (1 - 2^n*x)^(n+3).
%F A326012 E.g.f.: sum_{n>=0} ((n+1 + 2^n*x)*(n+2 + 2^n*x) + 2^n*x)/2 * 2^(n^2) * exp(2^n*x) * x^n/n!.
%e A326012 O.g.f.: A(x) = 1 + 9*x + 150*x^2 + 7290*x^3 + 1252815*x^4 + 821843253*x^5 + 2111728937500*x^6 + 21400822889676324*x^7 + 856401659982049536045*x^8 + ... + (n+1)*(n+2)/2 * (2^n + 1)^n*x^n + ...
%e A326012 such that
%e A326012 A(x) = 1/(1 - x)^3 + 3*2*x/(1 - 2*x)^4 + 6*2^4*x^2/(1 - 2^2*x)^5 + 10*2^9*x^3/(1 - 2^3*x)^6 + 15*2^16*x^4/(1 - 2^4*x)^7 + 21*2^25*x^5/(1 - 2^5*x)^8 + 28*2^36*x^6/(1 - 2^6*x)^9 + ... + (n+1)*(n+2)/2 * 2^(n^2)*x^n/(1 - 2^n*x)^(n+3) + ...
%t A326012 Table[((n+1)(n+2))/2 (2^n+1)^n,{n,0,20}] (* _Harvey P. Dale_, Jun 02 2025 *)
%o A326012 (PARI) {a(n) = (n+1)*(n+2)/2 * (2^n + 1)^n}
%o A326012 for(n=0,15, print1(a(n),", "))
%o A326012 (PARI) /* O.g.f. */
%o A326012 {a(n) = my(A = sum(m=0,n, (m+1)*(m+2)/2 * 2^(m^2) * x^m / (1 - 2^m*x +x*O(x^n))^(m+3) )); polcoeff(A,n)}
%o A326012 for(n=0,15, print1(a(n),", "))
%o A326012 (PARI) /* E.g.f. */
%o A326012 {a(n) = my(A = sum(m=0,n, ((m+1 + 2^m*x)*(m+2 + 2^m*x) + 2^m*x)/2 * 2^(m^2) * exp(2^m*x +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
%o A326012 for(n=0,15, print1(a(n),", "))
%Y A326012 Cf. A136516, A326011.
%K A326012 nonn
%O A326012 0,2
%A A326012 _Paul D. Hanna_, Jun 05 2019