OFFSET
0,5
COMMENTS
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
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.
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)*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)
EXAMPLE
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;
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Jun 15 2004
STATUS
approved