A216313 -id:A216313 - cp's OEIS frontend

A355267 Triangle read by rows, T(n, k) = n! * [y^k] [x^n] exp(1/(1 - x)^(1 + y) - 1), for 0 <= k <= n.

1, 1, 1, 3, 5, 2, 13, 29, 21, 5, 73, 200, 202, 90, 15, 501, 1609, 2045, 1295, 410, 52, 4051, 14809, 22418, 18085, 8220, 1998, 203, 37633, 153453, 267400, 259175, 151165, 53095, 10402, 877, 394353, 1767240, 3463612, 3889620, 2740885, 1241632, 353178, 57676, 4140

Offset: 0

Peter Luschny, Jul 05 2022

			Triangle T(n, k) begins:
[0]      1;
[1]      1,      1;
[2]      3,      5,      2;
[3]     13,     29,     21,      5;
[4]     73,    200,    202,     90,     15;
[5]    501,   1609,   2045,   1295,    410,    52;
[6]   4051,  14809,  22418,  18085,   8220,  1998,   203;
[7]  37633, 153453, 267400, 259175, 151165, 53095, 10402, 877;

Cf. A136658 (row sums), A000007 (alternating row sums), A000262 (column 0), A216313 (column 1), A000110 (main diagonal).

Cf. A355260.

egf := exp(1/(1 - x)^(1 + y) - 1):
ser := series(egf, x, 12): cfx := n -> coeff(ser, x, n):
seq(print(seq(n!*(coeff(cfx(n), y, k)), k = 0..n)), n = 0..8);

A306948 Expansion of e.g.f. (1 + x)log(1 + x)exp(x).

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 19, -15, 216, -1407, 11803, -108483, 1106192, -12363703, 150381243, -1977666743, 27965386320, -423158076351, 6822782712723, -116781368777867, 2114916140765496, -40404117909336247, 812091479233464131, -17130720178674680031, 378423227774537955688

Offset: 0

Ilya Gutkovskiy, Mar 17 2019

sign

Cf. A000110, A002741, A048994, A070071, A216313.

a:=series((1 + x)*log(1 + x)*exp(x),x=0,25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] BellB[k] k, {k, 0, n}], {n, 0, 24}]
Table[Sum[(-1)^(k - 1) Binomial[n, k] (n - k + 1) (k - 1)!, {k, 1, n}], {n, 0, 24}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(k)*k.
a(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)*(n - k + 1)*(k - 1)!.
a(n) ~ exp(-1) * (-1)^n * n! / n^2. - Vaclav Kotesovec, Mar 18 2019
Conjecture: D-finite with recurrence a(n) +(n-5)*a(n-1) +(-3*n+10)*a(n-2) +3*(n-3)*a(n-3) +(-n+3)*a(n-4)=0. - R. J. Mathar, Aug 20 2021

cp's OEIS Frontend

A355267 Triangle read by rows, T(n, k) = n! * [y^k] [x^n] exp(1/(1 - x)^(1 + y) - 1), for 0 <= k <= n.

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A306948 Expansion of e.g.f. (1 + x)log(1 + x)exp(x).

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cp's OEIS Frontend

A355267 Triangle read by rows, T(n, k) = n! * [y^k] [x^n] exp(1/(1 - x)^(1 + y) - 1), for 0 <= k <= n.

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Examples

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Maple

A306948 Expansion of e.g.f. (1 + x)*log(1 + x)*exp(x).

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Maple

Mathematica

Formula

A306948 Expansion of e.g.f. (1 + x)log(1 + x)exp(x).