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) ).
Original entry on oeis.org
1, -1, 37, -6833, 56377, -439772603, 27217772209, -202070742359, 80837575181815013, -155957202651688954367, 1770963292969902374951, -16092436217742770647634507, 2975968726866580246152132993, -963399772945511487665759472653, 3891037048609240492066339458106680163
Offset: 0
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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 *)
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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)])
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
T(n,1)*(1*n+1)! =
A129814(n) for all n>=0.
T(n,2)*(2*n+1)! =
A273198(n) for all n>=0.
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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 *)
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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)])
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