A091597 Triangle read by rows: T(n,0) = A001045(n+1), T(n,n)=1, T(n,m) = T(n-1,m-1) + T(n-1,m).

1, 1, 1, 3, 2, 1, 5, 5, 3, 1, 11, 10, 8, 4, 1, 21, 21, 18, 12, 5, 1, 43, 42, 39, 30, 17, 6, 1, 85, 85, 81, 69, 47, 23, 7, 1, 171, 170, 166, 150, 116, 70, 30, 8, 1, 341, 341, 336, 316, 266, 186, 100, 38, 9, 1, 683, 682, 677, 652, 582, 452, 286, 138, 47, 10, 1

Offset: 0

Author

Paul Barry, Jan 23 2004

Keywords

• easy
• nonn
• tabl

Comments

A Jacobsthal-Pascal triangle.

Equals triangle M * Pascal's triangle, M = an infinite lower triangular Toeplitz matrix with A078008: [1, 0, 2, 2, 6, 10, 22, 42, ...] in every column. - Gary W. Adamson, May 25 2009

Examples

			Triangle begins as:
    1;
    1,   1;
    3,   2,   1;
    5,   5,   3,   1;
   11,  10,   8,   4,   1;
   21,  21,  18,  12,   5,   1;
   43,  42,  39,  30,  17,   6,   1;
   85,  85,  81,  69,  47,  23,   7,  1;
  171, 170, 166, 150, 116,  70,  30,  8, 1;
  341, 341, 336, 316, 266, 186, 100, 38, 9, 1;
		

Links

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

Crossrefs

Columns include A001045, A000975, A011377.

Row sums are A059570.

Cf. A078008. - Gary W. Adamson, May 25 2009

Programs

GAP

Flat(List([0..12], n->List([0..n], k-> Sum([0..n], j-> 2^j*Binomial(n-j, k+j)) ))); # G. C. Greubel, Jun 04 2019

Magma

[[(&+[2^j*Binomial(n-j, k+j): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 04 2019

Maple

A091597 := proc(n,k)
    if k = 0 then
        A001045(n+1) ;
    elif k = n then
        1 ;
    elif k <0 or k > n then
        0 ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Oct 05 2012

Mathematica

Table[Sum[Binomial[n-j, k+j]*2^j, {j,0,n}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2019 *)

PARI

{T(n,k) = sum(j=0, n, 2^j*binomial(n-j, k+j))}; \\ G. C. Greubel, Jun 04 2019

Sage

[[sum(2^j*binomial(n-j, k+j) for j in (0..n)) for k in (0..n)] for n in [0..12]] # G. C. Greubel, Jun 04 2019

Formula

Number triangle: T(n, k) = Sum_{j=0..n} binomial(n-j, k+j)2^j.

Riordan array: (1/(1-x-2*x^2), x/(1-x)).

k-th column has g.f. (1/(1-x-2*x^2))*(x/(1-x))^k.

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - 2*T(n-3,k) - 2*T(n-3,k-1), T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=3, T(2,1)=2, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014