A319392 -id:A319392 - cp's OEIS frontend

A320032 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(-x)/(1 - k*x).

1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 5, 2, 1, 1, 3, 13, 29, 9, -1, 1, 4, 25, 116, 233, 44, 1, 1, 5, 41, 299, 1393, 2329, 265, -1, 1, 6, 61, 614, 4785, 20894, 27949, 1854, 1, 1, 7, 85, 1097, 12281, 95699, 376093, 391285, 14833, -1, 1, 8, 113, 1784, 26329, 307024, 2296777, 7897952, 6260561, 133496, 1

Offset: 0

Ilya Gutkovskiy, Oct 03 2018

For n > 0 and k > 0, A(n,k) gives the number of derangements of the generalized symmetric group S(k,n), which is the wreath product of Z_k by S_n. - Peter Kagey, Apr 07 2020

			E.g.f. of column k: A_k(x) = 1 + (k - 1)*x/1! + (2*k^2 - 2*k + 1)*x^2/2! + (6*k^3 - 6*k^2 + 3*k - 1)*x^3/3! + (24*k^4 - 24*k^3 + 12*k^2 - 4*k + 1)*x^4/4! + ...
Square array begins:
   1,   1,     1,      1,      1,       1,  ...
  -1,   0,     1,      2,      3,       4,  ...
   1,   1,     5,     13,     25,      41,  ...
  -1,   2,    29,    116,    299,     614,  ...
   1,   9,   233,   1393,   4785,   12281,  ...
  -1,  44,  2329,  20894,  95699,  307024,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Sami H. Assaf, Cyclic derangements, arXiv:1002.3138 [math.CO], 2010.
Hilarion L. M. Faliharimalala and Jiang Zeng, Derangements and Euler's difference table for C_l wr S_n, Electron. J. Combin. 15 (2008), #R65.
Wikipedia, Generalized symmetric group.

Columns k=0..5 give A033999, A000166, A000354, A000180, A001907, A001908.
Main diagonal gives A319392.
Cf. A320031.

A:= proc(n, k) option remember;
     `if`(n=0, 1, k*n*A(n-1, k)+(-1)^n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 07 2020

Table[Function[k, n! SeriesCoefficient[Exp[-x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, (-1)^n HypergeometricPFQ[{1, -n}, {}, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(-x)/(1 - k*x).
A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j!*k^j.
A(n,k) = (-1)^n*2F0(1,-n; ; k).

A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 6, 117, 4388, 266065, 23731314, 2923345621, 475364541672, 98623225721601, 25421365316232710, 7969388199705535141, 2985785305877403047820, 1317500933136749853197329, 676266417871227455138941242, 399516621958550611386236160405

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000166, A120485, A134095, A302397, A319392, A332408.

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(-x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 + k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A072364 = exp(-exp(-1)). - Vaclav Kotesovec, Jul 10 2021
E.g.f.: Sum_{k>=0} (k*x*exp(-x))^k. - Seiichi Manyama, Feb 19 2022

A330497 a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 0, 1, 26, 1089, 70124, 6495985, 821315214, 136115947009, 28651724077976, 7470040450004001, 2363470644596843330, 892244303052345224641, 396227360441775922668036, 204487588996059177697597969, 121370399839482643287189048374

Offset: 0

Ilya Gutkovskiy, Dec 18 2019

nonn

Cf. A009940, A025166, A061711, A277423, A307885, A318224, A319392, A330260.

[Factorial(n)*&+[(-1)^k*Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019

Join[{1}, Table[n! Sum[(-1)^k Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, 1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[-x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

a(n) = n! * [x^n] exp(-x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselJ(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

cp's OEIS Frontend

A320032 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(-x)/(1 - k*x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A330497 a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

Formula