A330603
a(n) = Sum_{k>=0} (k - n)^n / 2^(k + 1).
Original entry on oeis.org
1, 0, 3, -14, 155, -1834, 27867, -492246, 10068459, -232990178, 6025718963, -172182404734, 5387697769467, -183214963001082, 6728091949444491, -265348057242998822, 11185888456798395563, -501937946696294628946, 23886968118494957119011, -1201674025637823778926414
Offset: 0
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R:=PowerSeriesRing(Rationals(), 50);
A330603:= func< n | Coefficient(R!(Laplace( Exp(-n*x)/(2-Exp(x)) )), n) >;
[A330603(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024
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Table[Sum[(k - n)^n/2^(k + 1), {k, 0, Infinity}], {n, 0, 19}]
Table[HurwitzLerchPhi[1/2, -n, -n]/2, {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[-n x]/(2 - Exp[x]), {x, 0, n}], {n, 0, 19}]
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[factorial(n)*( exp(-n*x)/(2-exp(x)) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024
A292915
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563
Offset: 0
E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 3)*x^2/2! + (k^3 + 3*k^2 + 9*k + 13)*x^3/3! + (k^4 + 4*k^3 + 18*k^2 + 52*k + 75) x^4/4! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
3, 6, 11, 18, 27, 38, ...
13, 26, 51, 94, 161, 258, ...
75, 150, 299, 582, 1083, 1910, ...
541, 1082, 2163, 4294, 8345, 15666, ...
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R:=PowerSeriesRing(Rationals(), 50);
T:= func< n,k | Coefficient(R!(Laplace( Exp(k*x)/(2-Exp(x)) )), n) >;
A292915:= func< n,k | T(k,n-k) >;
[A292915(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024
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A:= proc(n, k) option remember; k^n +add(
binomial(n, j)*A(j, k), j=0..n-1)
end:
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 27 2017
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Table[Function[k, n! SeriesCoefficient[Exp[k x]/(2 - Exp[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, HurwitzLerchPhi[1/2, -n, k]/2][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
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a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n); \\ Seiichi Manyama, Dec 25 2023
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def T(n,k): return factorial(n)*( exp(k*x)/(2-exp(x)) ).series(x, n+1).list()[n]
def A292915(n,k): return T(k,n-k)
flatten([[A292915(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024
A337027
a(n) = (1/2) * Sum_{k>=0} (2*k + n)^n / 2^k.
Original entry on oeis.org
1, 3, 24, 293, 4784, 97687, 2393472, 68405073, 2233928448, 82063263371, 3349249267712, 150353137462717, 7362889615257600, 390601858379350815, 22315011551291080704, 1365896953310909493929, 89179296762081886011392, 6186383336743041502051219
Offset: 0
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Table[2^(n - 1) HurwitzLerchPhi[1/2, -n, n/2], {n, 0, 17}]
Table[n! SeriesCoefficient[Exp[n x]/(2 - Exp[2 x]), {x, 0, n}], {n, 0, 17}]
A340838
a(n) = (1/2) * Sum_{k>=0} (k*(k + n))^n / 2^k.
Original entry on oeis.org
1, 4, 139, 11928, 1909787, 491329088, 185373016419, 96425597012608, 66139668570414571, 57840395870803141632, 62813828698519808489915, 82933938539372018962724864, 130828514220436815006398809563, 243020960809424084526916839817216, 525038425527430196237626528753654867
Offset: 0
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Table[(1/2) Sum[(k (k + n))^n/2^k, {k, 0, Infinity}], {n, 0, 14}]
Join[{1}, Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, k - 2 n, 0] n^k, {k, 0, n}], {n, 1, 14}]]
Showing 1-4 of 4 results.
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