The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A326012 a(n) = (n+1)*(n+2)/2 * (2^n + 1)^n. 2
 1, 9, 150, 7290, 1252815, 821843253, 2111728937500, 21400822889676324, 856401659982049536045, 135337757314088315015913015, 84485579916959616279602050781250, 208476038116104541762965430756506783822, 2035321216424903698212658521136507048387952731, 78695103743782916150566765682074995597941860823900265, 12062337788631745341740759075565225314748425863190185546875000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS More generally, the following sums are equal: (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n, (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k), for any fixed integer k; here, k = 3 and p = 1, q = 2, r = x. LINKS FORMULA O.g.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (2^n + 1)^n * x^n. O.g.f.: Sum_{n>=0} (n+1)*(n+2)/2 * 2^(n^2) * x^n / (1 - 2^n*x)^(n+3). 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!. EXAMPLE 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 + ... such that 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) + ... PROG (PARI) {a(n) = (n+1)*(n+2)/2 * (2^n + 1)^n} for(n=0, 15, print1(a(n), ", ")) (PARI) /* O.g.f. */ {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)} for(n=0, 15, print1(a(n), ", ")) (PARI) /* E.g.f. */ {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)} for(n=0, 15, print1(a(n), ", ")) CROSSREFS Cf. A136516, A326011. Sequence in context: A296017 A218350 A027018 * A089916 A143000 A143905 Adjacent sequences:  A326009 A326010 A326011 * A326013 A326014 A326015 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 05 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 22 13:41 EST 2020. Contains 331149 sequences. (Running on oeis4.)