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.

A273198 a(n) = T(n,2) with T(n, m) = (m*n+1)! * Sum_{k=0..n}( 1/(m*k+1) * Sum_{j=0..m*k} (-1)^j*C(k,j)*j^(m*n) ).

Original entry on oeis.org

1, -2, 296, -327984, 1363872384, -15198541159680, 372495898187043840, -17616182020373076940800, 1464370216956293433318604800, -199499758936277018742988067635200, 42181903584776412718275835664105472000, -13251216132203374725100642797337549799424000
Offset: 0

Views

Author

Peter Luschny, Jun 26 2016

Keywords

Links

Crossrefs

Cf. A129814, A273196, A273197.

Programs

  • Mathematica
    Flatten[{1, Table[(2*n + 1)! * Sum[1/(2*k + 1)*Sum[(-1)^j*Binomial[k, j]*j^(2*n), {j, 0, 2*k}], {k, 0, n}], {n, 1, 10}]}] (* Vaclav Kotesovec, Jun 26 2016 *)
  • Sage
    def T(n, m): return factorial(m*n+1) * sum(1/(m*k+1)*sum((-1)^j*binomial(k,j)* j^(m*n) for j in (0..m*k)) for k in (0..n))
def a(n): return T(n, 2)
print([a(n) for n in (0..12)])

Formula

a(n) ~ (-1)^n * sqrt(Pi) * 2^(4*n) * n^(4*n + 1/2) / (sqrt(1-c) * exp(4*n) * c^n * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... . - Vaclav Kotesovec, Jun 26 2016

A273197 a(n) = denominator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(m*k+1) * Sum_{j=0..m*k} (-1)^j*C(k,j)*j^(m*n) ).

Original entry on oeis.org

1, 3, 15, 105, 15, 1155, 455, 15, 19635, 95095, 2145, 31395, 7735, 2805, 10818885, 50115065, 3315, 596505, 80925845, 3795, 18515805, 221847535, 2211105, 204920500785, 1453336885, 148335, 95055765, 287558635, 27897511785, 397299047145, 5613813089885, 8897205
Offset: 0

Views

Author

Peter Luschny, Jun 26 2016

Keywords

Comments

T(n,0) are the natural numbers, T(n,1) the Bernoulli numbers.

Crossrefs

Cf. A273196 (numerators).
T(n,0) = A000027(n) for n>=1.
T(n,1) = A027641(n)/A027642(n) for all n>=0.
T(n,1)*(1*n+1)! = A129814(n) for all n>=0.
T(n,2)*(2*n+1)! = A273198(n) for all n>=0.

Programs

  • Mathematica
    Table[Function[{n, m}, If[n == 0, 1, Denominator@ Sum[1/(m k + 1) Sum[(-1)^j Binomial[k, j] j^(m n), {j, 0, m k}], {k, 0, n}]]][n, 2], {n, 0, 31}] (* Michael De Vlieger, Jun 26 2016 *)
  • Sage
    def T(n, m): return sum(1/(m*k+1)*sum((-1)^j*binomial(k,j)*j^(m*n) for j in (0..m*k)) for k in (0..n))
def a(n): return T(n, 2).denominator()
print([a(n) for n in (0..31)])
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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