A292914 -id:A292914 - cp's OEIS frontend

A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - njx).

1, 1, 3, 19, 201, 3176, 69823, 2026249, 74565473, 3376695763, 183991725451, 11854772145800, 890415496931689, 77023751991841669, 7592990698770559111, 845240026276785888451, 105409073489605774592897, 14625467507717709778793020, 2244123413703647502288608467, 378751257186051653931253015229

Offset: 0

Ilya Gutkovskiy, Mar 20 2018

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..101
N. J. A. Sloane, Transforms

Cf. A000110, A004211, A004212, A004213, A005011, A005012, A008277, A075506, A075507, A075508, A075509, A242817, A292914, A318183.

List([0..20],n->Sum([0..n],k->n^(n-k)*Stirling2(n,k))); # Muniru A Asiru, Mar 20 2018

Table[SeriesCoefficient[Sum[x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(Exp[n x] - 1)/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[n^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 19}]]
(* Or: *)
A301419[n_] := If[n == 0, 1, n^n BellB[n, 1/n]];
Table[A301419[n], {n, 0, 19}] (* Peter Luschny, Dec 22 2021 *)

a(n) = sum(k=0, n, n^(n-k)*stirling(n, k, 2)); \\ Michel Marcus, Mar 23 2018

a(n) = n! * [x^n] exp((exp(n*x) - 1)/n), for n > 0.
a(n) = Sum_{k=0..n} n^(n-k)*Stirling2(n,k).
a(n) = n^n * BellPolynomial(n, 1/n) for n >= 1. - Peter Luschny, Dec 22 2021
a(n) ~ exp(n/LambertW(n^2) - n) * n^(2*n) / (sqrt(1 + LambertW(n^2)) * LambertW(n^2)^n). - Vaclav Kotesovec, Jun 06 2022

A307066 a(n) = exp(-1) * Sum_{k>=0} (n*k + 1)^n/k!.

Original entry on oeis.org

1, 2, 13, 199, 5329, 216151, 12211597, 909102342, 85761187393, 9957171535975, 1390946372509101, 229587693339867567, 44117901231194922193, 9748599124579281064294, 2451233017637221706477037, 695088863051920283838281851, 220558203335628758134165860609

Offset: 0

Ilya Gutkovskiy, Jun 24 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000110, A126390, A134980, A284859, A285064, A292914, A307080.

A307066:= func< n | (&+[Binomial(n,k)*n^k*Bell(k): k in [0..n]]) >;
[A307066(n): n in [0..31]]; // G. C. Greubel, Jan 24 2024

Table[Exp[-1] Sum[(n k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 16}]
Table[n! SeriesCoefficient[Exp[Exp[n x] + x - 1], {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[Binomial[n, k] n^k BellB[k], {k, 0, n}], {n, 1, 16}]]

def A307066(n): return sum(binomial(n,k)*n^k*bell_number(k) for k in range(n+1))
[A307066(n) for n in range(31)] # G. C. Greubel, Jan 24 2024

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

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

A330605 a(n) = exp(-1) * Sum_{k>=0} (n*k - 1)^n / k!.

Original entry on oeis.org

1, 0, 5, 89, 2737, 121399, 7316101, 572218716, 56142822849, 6731180810945, 965898950508901, 163116461798211503, 31969444766902475185, 7187057932197297484108, 1834860441330563739401765, 527403671798720265634312349, 169396494914472404237224898305

Offset: 0

Ilya Gutkovskiy, Dec 19 2019

nonn

Cf. A000110, A000296, A124311, A292914, A307066, A330604.

Table[Exp[-1] Sum[(n k - 1)^n/k!, {k, 0, Infinity}], {n, 0, 16}]
Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] n^k BellB[k], {k, 0, n}], {n, 1, 16}]]
Table[n! SeriesCoefficient[Exp[Exp[n x] - x - 1], {x, 0, n}], {n, 0, 16}]

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

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * n^k * Bell(k).

A295552 a(n) = n! * [x^n] exp(xexp(nx)).

Original entry on oeis.org

1, 1, 5, 46, 689, 15476, 483157, 19719022, 1009495489, 63119450152, 4728073048901, 417482964953594, 42834647403146161, 5043607239173464924, 674409403861210214485, 101517071981284179924526, 17074451852556909059698433, 3187883879639402167714593488, 656838643288782957496595002117

Offset: 0

Ilya Gutkovskiy, Nov 23 2017

nonn

N. J. A. Sloane, Transforms

Cf. A000248, A216689, A292914.

Table[n! SeriesCoefficient[Exp[x Exp[n x]], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[Sum[Binomial[n, k] (n k)^(n - k), {k, 0, n}], {n, 1, 18}]]

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

A292913 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(k*x)-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 15, 0, 1, 5, 32, 135, 240, 52, 0, 1, 6, 50, 320, 1215, 1664, 203, 0, 1, 7, 72, 625, 3840, 12636, 12992, 877, 0, 1, 8, 98, 1080, 9375, 53248, 147987, 112256, 4140, 0, 1, 9, 128, 1715, 19440, 162500, 831488, 1917999, 1059840, 21147, 0

Offset: 0

Ilya Gutkovskiy, Sep 26 2017

			E.g.f. of column k: A_k(x) =  1 + k*x/1! + 2*k^2*x^2/2! + 5*k^3*x^3/3! + 15*k^4 x^4/4! + 52*k^5*x^5/5! + ...
Square array begins:
1,   1,     1,      1,      1,       1,  ...
0,   1,     2,      3,      4,       5,  ...
0,   2,     8,     18,     32,      50,  ...
0,   5,    40,    135,    320,     625,  ...
0,  15,   240,   1215,   3840,    9375,  ...
0,  52,  1664,  12636,  53248,  162500,  ...

Columns k=0..3 give A000007, A000110, A055882, A247452.
Rows n=0..2 give A000012, A001477, A001105.
Main diagonal gives A292914.

A:= (n, k)-> k^n * combinat[bell](n):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 26 2017

Table[Function[k, n! SeriesCoefficient[Exp[Exp[k x] - 1], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-((-1)^(i + 1) (i - 1) + i + 3) k x/4, 1, {i, 0, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

O.g.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - 2*k*x/(1 - k*x/(1 - 3*k*x/(1 - k*x/(1 - 4*k*x/(1 - ...))))))))), a continued fraction.
E.g.f. of column k: exp(exp(k*x)-1).
A(n,k) = exp(-1)*k^n*Sum_{j>=0} j^n/j!.
A(n,k) = k^n * Bell(n). - Alois P. Heinz, Sep 26 2017

A308330 a(n) = n! * [x^n] exp(exp(n*x)/(1 - x) - 1).

Original entry on oeis.org

1, 2, 19, 346, 10217, 441226, 26023123, 1998840586, 193094418161, 22841006706928, 3239088790361491, 541309430523114804, 105106521730010262745, 23431755937256853296514, 5936989025261397848036755, 1694791457312643753292004446, 540937403928198054978670965089

Offset: 0

Ilya Gutkovskiy, May 20 2019

nonn

Cf. A063170, A292914, A308331, A321974.

Table[n! SeriesCoefficient[Exp[Exp[n x]/(1 - x) - 1], {x, 0, n}], {n, 0, 16}]

A308331 a(n) = n! * [x^n] exp(exp(n*x)/(1 + x) - 1).

Original entry on oeis.org

1, 0, 3, 50, 1449, 61724, 3608515, 275520972, 26505128433, 3125830471928, 442286373458691, 73789189395157730, 14309059313820886681, 3186711239965235356776, 806772967716453793227523, 230153293624841114893344854, 73420355768107554901016231265

Offset: 0

Ilya Gutkovskiy, May 20 2019

nonn

Cf. A134095, A292914, A308330, A321989.

Table[n! SeriesCoefficient[Exp[Exp[n x]/(1 + x) - 1], {x, 0, n}], {n, 0, 16}]

cp's OEIS Frontend

A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - n*j*x).

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GAP

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A307066 a(n) = exp(-1) * Sum_{k>=0} (n*k + 1)^n/k!.

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Magma

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A330605 a(n) = exp(-1) * Sum_{k>=0} (n*k - 1)^n / k!.

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Mathematica

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A295552 a(n) = n! * [x^n] exp(x*exp(n*x)).

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A292913 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(k*x)-1).

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Maple

Mathematica

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A308330 a(n) = n! * [x^n] exp(exp(n*x)/(1 - x) - 1).

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Mathematica

A308331 a(n) = n! * [x^n] exp(exp(n*x)/(1 + x) - 1).

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Mathematica

A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - njx).

A295552 a(n) = n! * [x^n] exp(xexp(nx)).