A273197 -id:A273197 - cp's OEIS frontend

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

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

Offset: 0

Peter Luschny, Jun 26 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..123

Cf. A129814, A273196, A273197.

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 *)

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)])

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

A273196 a(n) = numerator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).

Original entry on oeis.org

1, -1, 37, -6833, 56377, -439772603, 27217772209, -202070742359, 80837575181815013, -155957202651688954367, 1770963292969902374951, -16092436217742770647634507, 2975968726866580246152132993, -963399772945511487665759472653, 3891037048609240492066339458106680163

Offset: 0

Peter Luschny, Jun 26 2016

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

Cf. A273197 (denominator), T(n,0) = A000027, T(n,1) = A027641/A027642.

Also T(n,1)*(1*n+1)! = A129814, T(n,2)*(2*n+1)! = A273198.

Table[Function[{n, m}, If[n == 0, 1, Numerator@ 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, 14}] (* Michael De Vlieger, Jun 26 2016 *)

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).numerator()
print([a(n) for n in (0..14)])

cp's OEIS Frontend

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

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A273196 a(n) = numerator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).

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cp's OEIS Frontend

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) ).

Views

Author

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Programs

Mathematica

Sage

Formula

A273196 a(n) = numerator 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) ).

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Author

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Comments

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Programs

Mathematica

Sage

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

A273196 a(n) = numerator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).