A122101 T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A000670(n-k+i).

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, 47293, 42610, 38468, 34792, 31520, 28600, 25988, 23646, 545835, 498542, 455932, 417464, 382672, 351152, 322552, 296564, 272918

Offset: 0

Author

Vladeta Jovovic, Oct 18 2006

Keywords

• easy
• nonn
• tabl

Examples

			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;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Columns k=0-1 give: A000670, A232472.

Row sums give A089677(n+1).

Main diagonal gives A052841.

T(2n,n) gives A340837.

Cf. A005649, A069321, A073146.

Programs

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

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

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 *)

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

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)]

Formula

Doubly-exponential generating function: Sum_{n, k} a(n-k,k) x^n/n! y^k/k! = exp(-y)/(2-exp(x+y)).