A295183 -id:A295183 - cp's OEIS frontend

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

1, 0, 2, 6, 72, 620, 8640, 122346, 2156672, 41367672, 905126400, 21646532270, 570077595648, 16268377195044, 502096929431552, 16629319748711250, 588938142209310720, 22196966267762213744, 887352465220427317248, 37496112562144553167062, 1670071417348195942400000, 78195398849926292810318940

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

nonn

The n-th term of the n-fold exponential convolution of A000166 with themselves.

Robert Israel, Table of n, a(n) for n = 0..396
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Main diagonal of A295181.

Cf. A000166, A000407, A001813, A087981, A137775, A295183, A383379.

S:= series((exp(-x)/(1-x))^n,x,30):
seq(n!*coeff(S,x,n),n=0..29); # Robert Israel, Nov 16 2017

Table[n! SeriesCoefficient[Exp[-n x]/(1 - x)^n, {x, 0, n}], {n, 0, 21}]

a(n) = A295181(n,n).
a(n) ~ phi^(3*n - 1/2) * n^n / (5^(1/4) * exp(n*(1 + 1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017
a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k-1,k)/(n-k)!. - Seiichi Manyama, Apr 25 2025

A295188 Decimal expansion of phi^3 * exp(1 - 1/phi), where phi is the golden ratio.

Original entry on oeis.org

6, 2, 0, 6, 5, 2, 7, 0, 3, 8, 3, 9, 7, 1, 6, 3, 7, 3, 1, 0, 0, 0, 7, 4, 0, 5, 3, 2, 1, 8, 6, 5, 8, 0, 5, 8, 5, 2, 7, 8, 0, 5, 2, 8, 7, 0, 8, 4, 7, 9, 6, 2, 0, 2, 2, 9, 2, 6, 0, 7, 5, 3, 9, 6, 8, 7, 9, 0, 5, 8, 4, 9, 3, 7, 5, 6, 1, 4, 1, 8, 4, 4, 4, 3, 5, 6, 3, 1, 1, 2, 2, 6, 1, 0, 2, 3, 0, 5, 0, 6, 3, 7, 0, 2, 4

Offset: 1

Vaclav Kotesovec, Nov 16 2017

			6.206527038397163731000740532186580585278052870847962022926...

Index entries for transcendental numbers

Cf. A001622, A066399, A239761, A295183.

evalf(((1+sqrt(5))/2)^3 * exp(1 - 2/(1+sqrt(5))), 120);

RealDigits[GoldenRatio^3 * Exp[1 - 1/GoldenRatio], 10, 110][[1]]

phi=(sqrt(5)+1)/2; phi^3*exp(2-phi) \\ Charles R Greathouse IV, Nov 21 2024

Equals ((1+sqrt(5))/2)^3 * exp(1 - 2/(1+sqrt(5))).
Equals limit n->infinity (A066399(n)/n!)^(1/n).
Equals limit n->infinity (A239761(n)/n!)^(1/n).
Equals limit n->infinity (A295183(n)/n!)^(1/n).

cp's OEIS Frontend

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

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