OFFSET
0,8
LINKS
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.
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, ...
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
KEYWORD
AUTHOR
Peter Luschny, Mar 24 2015
STATUS
approved