cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Peter Luschny, Mar 24 2015

Keywords

Examples

			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;

Links

Crossrefs

Main diagonal gives A293145.
Cf. A000262, A001477, A005563, A052897, A226514, A255806, A256325.

Programs

  • Magma
    [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
  • Maple
    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;
  • Mathematica
    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 *)
  • PARI
    {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
  • PARI
    {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
  • Sage
    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

Formula

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)

A317364 Expansion of e.g.f. exp(2*x/(1 + x)).

Original entry on oeis.org

1, 2, 0, -4, 16, -48, 64, 800, -12288, 127232, -1150976, 9266688, -58726400, 68777984, 7510646784, -207794409472, 4241007640576, -77359570944000, 1321952191971328, -21274345818161152, 313768799799607296, -3838962981483839488, 21775623343518515200, 859024717017756205056
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2018

Keywords

Comments

Inverse Lah transform of the powers of 2 (A000079).

Links

Crossrefs

Cf. A000079, A052897, A111884.
Cf. A001263, A008297, A086915.

Programs

  • Magma
    [n eq 0 select 1 else (-1)^n*Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), 2): n in [0..25]]; // G. C. Greubel, Feb 23 2021
  • Maple
    a:= proc(n) option remember; add((-1)^(n-k)*
      n!/k!*binomial(n-1, k-1)*2^k, k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[2 x/(1 + x)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[Exp[-2 (-x)^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] 2^k n!/k!, {k, 0, n}], {n, 0, 23}]
Join[{1}, Table[2 (-1)^(n+1) n! Hypergeometric1F1[1-n, 2, 2], {n, 23}]]
  • PARI
    a(n) = if (n==0, 1, (-1)^n*n!*pollaguerre(n, -1, 2)); \\ Michel Marcus, Feb 23 2021
  • Sage
    [1 if n==0 else (-1)^n*factorial(n)*gen_laguerre(n, -1, 2) for n in (0..25)] # G. C. Greubel, Feb 23 2021

Formula

E.g.f.: Product_{k>=1} exp(-2*(-x)^k).
a(n) = 2*(-1)^(n+1) * n! * Hypergeometric1F1([1-n], [2], 2).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*2^k*n!/k!.
(n^2 + n)*a(n) + 2*n*a(n+1) + a(n+2) = 0. - Robert Israel, Aug 18 2019
From G. C. Greubel, Feb 23 2021: (Start)
a(n) = (-1)^n * n! * Laguerre(n, -1, 2) for n > 0 with a(0) = 1.
a(n) = Sum_{k=0..n} (-1)^(n-k) * A086915(n, k).
a(n) = (-1)^n * Sum_{k=0..n} 2^k * A008297(n, k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (n-k+1)! * A001263(n, k). (End)
Showing 1-2 of 2 results.

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