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, 125853, 606417, 1809905, 3197210, 2995011, 1112083, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

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

Examples

			Triangle starts:
  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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• 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.

Crossrefs

Diagonals include: A000012, A000217, A002104.

Sums include: A000522 (row), A001339, A023443 (alternating sign row), A082030, A081367.

Programs

Magma

A046716:= func< n,k | (&+[(-1)^j*Binomial(n,k-j)*StirlingFirst(j+n-k, n-k): j in [0..k]]) >;
[A046716(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 31 2024

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 *)
T[n_, k_]:= T[n,k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, T[n-1,k] +n*T[n-1,k-1] - (n-1)*T[n-2,k-2]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 31 2024 *)

SageMath

def A046716(n, k): return sum(binomial(n, k-j)*stirling_number1(j+n-k, n-k) for j in range(k+1))
flatten([[A046716(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 31 2024

Formula

Enneking and Ahuja reference gives the recurrence t(n, k) = t(n-1, k) - n*t(n-1, k-1) - (n-1)*t(n-2, k-2), with t(n, 0) = 1 and t(n, n) = (-1)^n. This sequence is T(n, k) = (-1)^k * t(n, k).

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).

From G. C. Greubel, Jul 31 2024: (Start)

T(n, k) = T(n-1, k) + n*T(n-1, k-1) - (n-1)*T(n-2, k-2), with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^(n+1)*A023443(n). (End)

Extensions

More terms from Vladeta Jovovic, Jun 15 2004