A194583 Triangle T(n,k) with T(n,0)=1 and T(n,k) = (2^(n+1)-2^k)*T(n,k-1) + T(n+1,k-1) otherwise.

1, 1, 3, 1, 7, 43, 1, 15, 211, 2619, 1, 31, 931, 26251, 654811, 1, 63, 3907, 234795, 13255291, 662827803, 1, 127, 16003, 1985131, 238658491, 26961325147, 2699483026843, 1, 255, 64771, 16323819, 4050110011, 973958217435, 220115609012251, 44102911693372059, 1, 511, 260611, 132393451, 66733574971, 33115631264731, 15928113739803931, 7200501591899676571, 2886238576935227688091

Offset: 0

Author

R. J. Mathar, Aug 29 2011

Keywords

• nonn
• tabl

Examples

			The triangle starts in row n=0 as
  1;
  1,  3;
  1,  7,  43;
  1, 15, 211,  2619;
  1, 31, 931, 26251, 654811;
		

Links

• G. Helms, Number array not found in OEIS, SeqFan list Aug 27 2011

Programs

Maple

A194583 := proc(n,k) option remember; if n=0 or k=0 then 1; elif k> n then
return procname(k,n); else (2^(n+1)-2^k)*procname(n,k-1)+procname(n+1,k-1) ; end if;
end proc:

Mathematica

t[, 0] = 1; t[n, k_] := t[n, k] = (2^(n+1)-2^k)*t[n, k-1]+t[n+1, k-1]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014 *)

Formula

T(n,1) = A000225(n+1).

T(n,2) = (2^(n+1)-4)*(2^(n+1)-1) + 2^(n+2) - 1.

T(n,k) = -Sum_{j=1..k+1} A158474(k+1,j)*T(n-j,k) assuming the symmetric extension T(n,k) = T(k,n).