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

0,8

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Index entries for sequences related to Laguerre polynomials

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Main diagonal gives A293145.

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

Sequence in context: A370506 A184182 A085771 * A344499 A284799 A111106

Adjacent sequences: A253283 A253284 A253285 * A253287 A253288 A253289

Keyword

tabl,easy,nonn

Author

Peter Luschny, Mar 24 2015

Status

approved