A293145 -id:A293145 - cp's OEIS frontend

A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!C(n,n-j) C(n-1,n-j)*k^j, for n>=0 and k>=0.

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 8, 3, 1, 0, 73, 44, 15, 4, 1, 0, 501, 304, 99, 24, 5, 1, 0, 4051, 2512, 801, 184, 35, 6, 1, 0, 37633, 24064, 7623, 1696, 305, 48, 7, 1, 0, 394353, 261536, 83079, 18144, 3145, 468, 63, 8, 1

Offset: 0

Peter Luschny, Mar 24 2015

			Square array starts, A(n,k):
      1,       1,       1,       1,      1,      1,      1, ...  A000012
      0,       1,       2,       3,      4,      5,      6, ...  A001477
      0,       3,       8,      15,     24,     35,     48, ...  A005563
      0,      13,      44,      99,    184,    305,    468, ...  A226514
      0,      73,     304,     801,   1696,   3145,   5328, ...
      0,     501,    2512,    7623,  18144,  37225,  68976, ...
      0,    4051,   24064,   83079, 220096, 495475, 997056, ...
A000007, A000262, A052897, A255806, ...
Triangle starts, T(n, k) = A(n-k, k):
  1;
  0,   1;
  0,   1,   1;
  0,   3,   2,  1;
  0,  13,   8,  3,  1;
  0,  73,  44, 15,  4, 1;
  0, 501, 304, 99, 24, 5, 1;

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Laguerre polynomials

Main diagonal gives A293145.

Cf. A000262, A001477, A005563, A052897, A226514, A255806, A256325.

[k eq n select 1 else k*Factorial(n-k-1)*Evaluate(LaguerrePolynomial(n-k-1, 1), -k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

L := (n, k) -> (n-k)!*binomial(n,n-k)*binomial(n-1,n-k):
A := (n, k) -> add(L(n,j)*k^j, j=0..n):
# Alternatively:
# A := (n, k) -> `if`(n=0,1, simplify(k*n!*hypergeom([1-n],[2],-k))):
for n from 0 to 6 do lprint(seq(A(n,k), k=0..6)) od;

A253286[n_, k_]:= If[k==n, 1, k*(n-k-1)!*LaguerreL[n-k-1, 1, -k]];
Table[A253286[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

{T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^j*binomial(n-1, j-1)/j!))} \\ Seiichi Manyama, Feb 03 2021

{T(n, k) = if(n<2, (k-1)*n+1, (2*n+k-2)*T(n-1, k)-(n-1)*(n-2)*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021

flatten([[1 if k==n else k*factorial(n-k-1)*gen_laguerre(n-k-1, 1, -k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

A(n,k) = k*n!*hypergeom([1-n],[2],-k) for n>=1 and 1 for n=0.
Row sums of triangle, Sum_{k=0..n} A(n-k, k) = 1 + A256325(n).
From Seiichi Manyama, Feb 03 2021: (Start)
E.g.f. of column k: exp(k*x/(1-x)).
T(n,k) = (2*n+k-2) * T(n-1,k) - (n-1) * (n-2) * T(n-2, k) for n > 1. (End)
From G. C. Greubel, Feb 23 2021: (Start)
A(n, k) = k*(n-1)!*LaguerreL(n-1, 1, -k) with A(0, k) = 1.
T(n, k) = k*(n-k-1)!*LaguerreL(n-k-1, 1, -k) with T(n, n) = 1.
T(n, 2) = A052897(n) = A086915(n)/2.
Sum_{k=0..n} T(n, k) = 1 + Sum_{k=0..n-1} (n-k-1)*k!*LaguerreL(k, 1, k-n+1). (End)

A293146 a(n) = n! * [x^n] exp(x/(1 - n*x)).

Original entry on oeis.org

1, 1, 5, 73, 2161, 108101, 8201701, 878797165, 126422091713, 23514740267401, 5492576235204901, 1574136880033408241, 543143967119720304625, 222106209904092987888013, 106221716052645457812866501, 58741017143127754662557082901, 37194600833984874761008613195521

Offset: 0

Ilya Gutkovskiy, Oct 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..232

Cf. A000262, A025168, A293145.

S:=series(exp(x/(1-n*x)),x,31):
seq(coeff(S,x,n)*n!,n=0..30); # Robert Israel, Oct 01 2017

Table[n! SeriesCoefficient[Exp[x/(1 - n x)], {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[Product[Exp[n^k x^(k + 1)], {k, 0, n}], {x, 0, n}], {n, 1, 16}]]
Join[{1}, Table[Sum[n^(n - k) n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 16}]]
Join[{1}, Table[n^n (n - 1)! Hypergeometric1F1[1 - n, 2, -1/n], {n, 1, 16}]]

{a(n) = if(n==0, 1, n!*sum(k=1, n, n^(n-k)*binomial(n-1, k-1)/k!))} \\ Seiichi Manyama, Feb 03 2021

a(n) ~ BesselI(1, 2) * sqrt(2*Pi) * n^(2*n-1/2) / exp(n). - Vaclav Kotesovec, Oct 01 2017

a(n) = n! * Sum_{k=1..n} n^(n-k) * binomial(n-1,k-1)/k! for n > 0. - Seiichi Manyama, Feb 03 2021

A317279 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n-1,k-1)n^k*n!/k!.

Original entry on oeis.org

1, 1, 0, -9, -32, 225, 3456, 2695, -433152, -4495743, 47872000, 1768142871, 6703534080, -597265448351, -11959736205312, 126058380654375, 9454322092343296, 84694164336894465, -5776865438988238848, -192541299662555831753, 1511905067561779200000, 243338391925401706938081, 3972949090873574466519040

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

sign

a(n) is the n-th term of the inverse Lah transform of the powers of n.

G. C. Greubel, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Transforms
Index entries for sequences related to Laguerre polynomials

Cf. A111884, A293145, A317277, A317278.

l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
[1]cat[(-1)^(n+1)*Factorial(n)*l(n-1,1,n): n in [1..30]]; // G. C. Greubel, Mar 09 2021

A317279:= n -> `if`(n=0,1,(-1)^(n+1)*n!*simplify(LaguerreL(n-1,1,n), 'LaguerreL'));
seq(A317279(n), n = 0..30); # G. C. Greubel, Mar 09 2021

Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] n^k n!/k!, {k, n}], {n, 22}]]
Table[n! SeriesCoefficient[Exp[n x/(1 + x)], {x, 0, n}], {n, 0, 22}]
Table[n! SeriesCoefficient[Product[Exp[-n (-x)^k], {k, n}], {x, 0, n}], {n, 0, 22}]
Join[{1}, Table[(-1)^(n+1) n n! Hypergeometric1F1[1-n, 2, n], {n, 22}]]

a(n) = if (n==0, 1, (-1)^(n+1)*n!*pollaguerre(n-1, 1, n)); \\ Michel Marcus, Mar 10 2021

[1]+[(-1)^(n+1)*factorial(n)*gen_laguerre(n-1,1,n) for n in (1..30)] # G. C. Greubel, Mar 09 2021

a(n) = n! * [x^n] exp(n*x/(1 + x)).
a(n) = n! * [x^n] Product_{k>=1} exp(-n*(-x)^k).
a(n) = (-1)^(n+1) * n * n! * Hypergeometric1F1([1-n], [2], n) with a(0) = 1.
a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1, n) with a(0) = 1. - G. C. Greubel, Mar 09 2021

A317277 a(n) = Sum_{k=0..n} binomial(n-1,k-1)k^nn!/k!; a(0) = 1.

Original entry on oeis.org

1, 1, 6, 81, 1828, 60565, 2734926, 160109005, 11724156648, 1045312448841, 111114793839610, 13845807451708441, 1994597720747571468, 328351264019737949341, 61162428777982281583302, 12782305566531823350524805, 2975150384583838798131401296, 766253903501365584725344992529

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

nonn

a(n) is the n-th term of the Lah transform of the n-th powers.

G. C. Greubel, Table of n, a(n) for n = 0..250
N. J. A. Sloane, Transforms

Cf. A000262, A052852, A103194, A293145, A317278, A317279.

[1]cat[(&+[Binomial(n-1,j-1)*Binomial(n,j)*Factorial(n-j)*j^n: j in [0..n]]): n in [1..30]]; // G. C. Greubel, Mar 09 2021

A317277:= n-> `if`(n=0,1, add(binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n)); seq(A317277(n), n=0..30); # G. C. Greubel, Mar 09 2021

Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^n n!/k!, {k, n}], {n, 17}]]
Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 - x))^k/k!, {k, n}], {x, 0, n}], {n, 17}]]

a(n) = if (n==0, 1, sum(k=0, n, binomial(n-1, k-1)*k^n*n!/k!)); \\ Michel Marcus, Mar 10 2021; corrected Jun 15 2022

[1]+[sum(binomial(n-1,j-1)*binomial(n,j)*factorial(n-j)*j^n for j in (0..n)) for n in (1..30)] # G. C. Greubel, Mar 09 2021

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

Name edited by Michel Marcus, Jun 15 2022

cp's OEIS Frontend

A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!*C(n,n-j)* C(n-1,n-j)*k^j, for n>=0 and k>=0.

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A293146 a(n) = n! * [x^n] exp(x/(1 - n*x)).

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A317279 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*n^k*n!/k!.

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A317277 a(n) = Sum_{k=0..n} binomial(n-1,k-1)*k^n*n!/k!; a(0) = 1.

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A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!C(n,n-j) C(n-1,n-j)*k^j, for n>=0 and k>=0.

A317279 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n-1,k-1)n^k*n!/k!.

A317277 a(n) = Sum_{k=0..n} binomial(n-1,k-1)k^nn!/k!; a(0) = 1.