OFFSET
0,4
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
Doubly-exponential generating function: Sum_{n, k} a(n-k,k) x^n/n! y^k/k! = exp(-y)/(2-exp(x+y)).
EXAMPLE
Triangle begins as:
1;
1, 0;
3, 2, 2;
13, 10, 8, 6;
75, 62, 52, 44, 38;
541, 466, 404, 352, 308, 270;
4683, 4142, 3676, 3272, 2920, 2612, 2342;
...
MAPLE
T:= (n, k)-> k!*(n-k)!*coeff(series(coeff(series(exp(-y)/
(2-exp(x+y)), y, k+1), y, k), x, n-k+1), x, n-k):
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Oct 02 2019
# second Maple program:
b:= proc(n) option remember; `if`(n<2, 1,
add(b(n-j)*binomial(n, j), j=1..n))
end:
T:= (n, k)-> add(binomial(k, j)*(-1)^j*b(n-j), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Oct 02 2019
MATHEMATICA
A000670[n_]:= If[n==0, 1, Sum[k! StirlingS2[n, k], {k, n}]]; T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[k, j]*A000670[n-k+j], {j, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 02 2019 *)
PROG
(PARI)
A000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
T(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(k, j)*A000670(n-k+j));
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 02 2019
(MAGMA)
A000670:= func< n | &+[Factorial(k)*StirlingSecond(n, k): k in [0..n]] >;
[(&+[(-1)^(k-j)*Binomial(k, j)*A000670(n-k+j): j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 02 2019
(Sage)
def A000670(n): return sum(factorial(k)*stirling_number2(n, k) for k in (0..n))
def T(n, k): return sum((-1)^(k-j)*binomial(k, j)*A000670(n-k+j) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)]
(GAP)
A000670:= function(n)
return Sum([0..n], i-> Factorial(i)*Stirling2(n, i) ); end;
T:= function(n, k)
return Sum([0..k], j-> (-1)^(k-j)*Binomial(k, j)*A000670(n-k+j) ); end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Oct 02 2019
CROSSREFS
KEYWORD
AUTHOR
Vladeta Jovovic, Oct 18 2006
STATUS
editing