cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A136516 a(n) = (2^n+1)^n.

Original entry on oeis.org

1, 3, 25, 729, 83521, 39135393, 75418890625, 594467302491009, 19031147999601100801, 2460686496619787545743873, 1280084544196357822418212890625, 2672769719437237714909813214827010049, 22366167213460480200139104627873703828439041
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2008

Keywords

Comments

More generally, Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! = Sum_{n>=0} (m*q^n + b)^n * x^n / n! for all q, m, b.
Main diagonal of A264871. - Omar E. Pol, Nov 27 2015

Examples

			A(x) = 1 + 3x + 5^2*x^2/2! + 9^3*x^3/3! + 17^4*x^4/4! +... + (2^n+1)^n*x^n/n! +...
A(x) = exp(x) + 2*exp(2x) + 2^4*exp(4x)*x^2/2! + 2^9*exp(8x)*x^3/3! +...+ 2^(n^2)*exp(2^n*x)*x^n/n! +...
This is a special case of the more general statement:
Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1, b=1.

Links

Crossrefs

Cf. A014050, A055601, A243918, A202989, A326011, A326012.

Programs

  • Magma
    [(2^n+1)^n: n in [0..45]]; // Vincenzo Librandi, Apr 21 2011
  • Maple
    seq((2^n+1)^n, n=0..30); # Robert Israel, Nov 27 2015
  • Mathematica
    Table[(2^n+1)^n,{n,0,16}] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
  • PARI
    a(n)=polcoeff(sum(k=0,n,2^(k^2)*exp(2^k*x)*x^k/k!),n)
  • PARI
    {a(n)=polcoeff(sum(k=0, n, 2^(k^2)*x^k/(1-2^k*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Sep 15 2009

Formula

E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * exp(2^n*x) * x^n/n!.
O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 - 2^n*x)^(n+1) = Sum_{n>=0} (2^n+1)^n*x^n. [Paul D. Hanna, Sep 15 2009]
a(n) = 2^(n^2) + n 2^(n^2-n) + O(n^2 2^(n^2-2n)). - Robert Israel, Nov 27 2015

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

Original entry on oeis.org

1, 9, 150, 7290, 1252815, 821843253, 2111728937500, 21400822889676324, 856401659982049536045, 135337757314088315015913015, 84485579916959616279602050781250, 208476038116104541762965430756506783822, 2035321216424903698212658521136507048387952731, 78695103743782916150566765682074995597941860823900265, 12062337788631745341740759075565225314748425863190185546875000
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2019

Keywords

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.

Examples

			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) + ...

Crossrefs

Cf. A136516, A326011.

Programs

  • Mathematica
    Table[((n+1)(n+2))/2 (2^n+1)^n,{n,0,20}] (* Harvey P. Dale, Jun 02 2025 *)
  • 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),", "))

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!.
Showing 1-2 of 2 results.

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