A046716 Coefficients of a special case of Poisson-Charlier polynomials.

1, 1, 1, 1, 3, 1, 1, 6, 8, 1, 1, 10, 29, 24, 1, 1, 15, 75, 145, 89, 1, 1, 21, 160, 545, 814, 415, 1, 1, 28, 301, 1575, 4179, 5243, 2372, 1, 1, 36, 518, 3836, 15659, 34860, 38618, 16072, 1, 1, 45, 834, 8274, 47775, 163191, 318926, 321690, 125673, 1, 1, 55, 1275, 16290

Offset

0,5

Comments

Diagonals: A000012, A000217; A000012, A002104. - Philippe Deléham, Jun 12 2004

The sequence a(n) = Sum_{k = 0..n} T(n,k)*x^(n-k) is the binomial transform of the sequence b(n) = (n+x-1)! / (x-1)!. - Philippe Deléham, Jun 18 2004

Links

Table of n, a(n) for n=0..58.

E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.

C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

Formula

Reference gives a recurrence.

Sum_{k = 0..n} T(n, k)*x^(n-k) = A000522(n), A001339(n), A082030(n) for x = 1, 2, 3 respectively. Sum_{k = 0..n} T(n, k)*2^k = A081367(n). - Philippe Deléham, Jun 12 2004

Let P(x, n) = Sum_{k = 0..n} T(n, k)*x^k, then Sum_{n>=0} P(x, n)*t^n / n! = exp(xt)/(1-xt)^(1/x). - Philippe Deléham, Jun 12 2004

T(n, 0) = 1, T(n, k) = (-1)^k * Sum_{i=n-k..n} (-1)^i*C(n, i)*S1(i, n-k), where S1 = Stirling numbers of first kind (A008275).

Example

Triangle starts:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  8,  1;
  1, 10, 29, 24,  1;
  ...

Maple

a := proc(n, k) option remember;
   if k = 0 then 1
elif k < 0 then 0
elif k = n then (-1)^n
else a(n-1, k) - n*a(n-1, k-1) - (n-1)*a(n-2, k-2) fi end:
A046716 := (n, k) -> abs(a(n, k));
seq(seq(A046716(n, k), k=0..n), n=0..9); # Peter Luschny, Apr 05 2011

Mathematica

t[_, 0] = 1; t[n_, k_] := (-1)^k*Sum[(-1)^i*Binomial[n, i]*StirlingS1[i, n-k], {i, n-k, n}]; Table[t[n, k] // Abs, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014 *)

Crossrefs

Sequence in context: A137251 A370757 A158359 * A371967 A202605 A298636

Adjacent sequences: A046713 A046714 A046715 * A046717 A046718 A046719

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jun 15 2004

Status

approved