A291980 -id:A291980 - cp's OEIS frontend

A291981 a(n) = n! * [x^n] exp(x)/(1 - log(1+x)).

1, 2, 4, 9, 23, 70, 244, 1015, 4605, 25090, 137516, 963589, 5832243, 54685670, 310755964, 4567695283, 15162041017, 581160322338, -1133608829900, 115627157052929, -1056546484541681, 34518230044201062, -541838359813821916, 14135169210980556655, -288911084512245360587

Offset: 0

Peter Luschny, Sep 16 2017

sign

Row sums of A291980.

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A291980.

a_list := proc(n) exp(x)/(1 - log(1+x)): series(%, x, n+1):
seq(j!*coeff(%, x, j), j=0..n) end: a_list(24);

nmax = 20; CoefficientList[Series[E^x/(1 - Log[1+x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 18 2017 *)

x='x+O('x^30); Vec(serlaplace(exp(x)/(1 - log(1+x)))) \\ G. C. Greubel, Aug 30 2018

a(n) ~ n! * (-1)^(n+1) * exp(-1) / (log(n)^2 * n) * (1 - 2*(1 + gamma)/log(n)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 18 2017

a(n) = 1 + Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 28 2021

A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)n![t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 9, 3, 1, 88, 56, 18, 4, 1, 694, 440, 140, 30, 5, 1, 6578, 4164, 1320, 280, 45, 6, 1, 72792, 46046, 14574, 3080, 490, 63, 7, 1, 920904, 582336, 184184, 38864, 6160, 784, 84, 8, 1, 13109088, 8288136, 2620512, 552552, 87444, 11088, 1176, 108, 9, 1

Offset: 0

Peter Luschny, Sep 15 2017

			Triangle starts:
[1]
[1,           1]
[3,           2,      1]
[14,          9,      3,     1]
[88,         56,     18,     4,    1]
[694,       440,    140,    30,    5,   1]
[6578,     4164,   1320,   280,   45,   6,  1]
[72792,   46046,  14574,  3080,  490,  63,  7, 1]
[920904, 582336, 184184, 38864, 6160, 784, 84, 8, 1]

Row sums: A291978.
Columns: A007840 (c=1), A052860 (c=2).
Diagonal: A045943 (d=3).
Cf. A291980.

T_row := proc(n) exp(x*t)/(1 + log(1+x)): series(%, x, n+1):
seq(coeff((-1)^(n-k)*n!*coeff(%,x,n),t,k), k=0..n) end:
seq(T_row(n), n=0..9);

T[n_, k_] := Binomial[n, n - k]*Sum[j!*Abs[StirlingS1[n - k, j]], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, May 12 2024 *)

T(n, k) = binomial(n, n - k)*Sum_{j=0..n - k} j!*abs(Stirling1(n - k, j)). - Detlef Meya, May 12 2024

cp's OEIS Frontend

A291981 a(n) = n! * [x^n] exp(x)/(1 - log(1+x)).

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A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)n![t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

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cp's OEIS Frontend

A291981 a(n) = n! * [x^n] exp(x)/(1 - log(1+x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)*n!*[t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)n![t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.