A340822 -id:A340822 - cp's OEIS frontend

A340823 a(n) = exp(-1) * Sum_{k>=0} (k*(k - n))^n / k!.

1, 1, 3, 5, 124, -2075, 91993, -4709903, 312334595, -25531783799, 2524083665172, -296260739274275, 40667620527027177, -6446882734412545043, 1167717545574222779643, -239452569059443831797303, 55146244227862697483251020, -14163492441645773105212592623

Offset: 0

Ilya Gutkovskiy, Jan 22 2021

sign

G. C. Greubel, Table of n, a(n) for n = 0..260

Cf. A000110, A020556, A020557, A290219, A334243, A340822.

A340823:= func< n | (&+[(-n)^j*Binomial(n,j)*Bell(2*n-j): j in [0..n]]) >;
[A340823(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

Table[Exp[-1] Sum[(k (k - n))^n/k!, {k, 0, Infinity}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[2 n - k] (-n)^k, {k, 0, n}], {n, 1, 17}]]

def A340823(n): return sum( binomial(n,k)*bell_number(2*n-k)*(-n)^k for k in range(n+1))
[A340823(n) for n in range(31)] # G. C. Greubel, Jun 12 2024

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(2*n-k) * (-n)^k.

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

Ilya Gutkovskiy, Jan 23 2021

nonn

Cf. A000670, A098696, A249938, A292916, A340822.

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}]]

a(n) = Sum_{k=0..n} binomial(n,k) * A000670(2*n-k) * n^k.

cp's OEIS Frontend

A340823 a(n) = exp(-1) * Sum_{k>=0} (k*(k - n))^n / k!.

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