A130265 Triangle read by rows: matrix product A007318 * A051340.

1, 2, 2, 4, 5, 3, 8, 10, 10, 4, 16, 19, 23, 17, 5, 32, 36, 46, 46, 26, 6, 64, 69, 87, 102, 82, 37, 7, 128, 134, 162, 204, 204, 134, 50, 8, 256, 263, 303, 387, 443, 373, 205, 65, 9, 512, 520, 574, 718, 886, 886, 634, 298, 82, 10

Offset: 0

Author

Gary W. Adamson, May 18 2007

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle are:
   1;
   2,  2;
   4,  5,  3;
   8, 10, 10,   4;
  16, 19, 23,  17,  5;
  32, 36, 46,  46, 26,  6;
  64, 69, 87, 102, 82, 37,  7;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A001787 (row sums), A007318, A051340, A058877, A084633, A258431.

Programs

Magma

A130265:= func< n,k | k eq n select n+1 else (k+1)*Binomial(n,k) + (&+[Binomial(n, j+k): j in [1..n-k]]) >;
[A130265(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2023

Maple

A051340 := proc(n,k)
    if k = n then
        n+1 ;
    elif k <= n then
        1;
    else
        0;
    end if;
end proc:
A130265 := proc(n,k)
    add( binomial(n,j)*A051340(j,k),j=k..n) ;
end proc:
seq(seq(A130265(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 06 2016

Mathematica

T[n_, k_]:= (k+1)*Binomial[n,k] + Binomial[n,k+1]*Hypergeometric2F1[1, k-n+1, k+2, -1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2023 *)

SageMath

def A130265(n,k): return (k+1)*binomial(n,k) + sum(binomial(n, j+k) for j in range(1,n-k+1))
flatten([[A130265(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 18 2023

Formula

Binomial transform of A051340.

From G. C. Greubel, Mar 18 2023: (Start)

T(n, k) = (k+1)*binomial(n,k) + Sum_{j=1..n-k} binomial(n, j+k).

T(n, k) = (k+1)*binomial(n,k) + binomial(n,k+1)*Hypergeometric2F1([1, k-n+1], [k+2], -1).

T(2*n, n) = (1/2)*T(2*n+1, n) = A258431(n+1).

Sum_{k=0..n} T(n, k) = A001787(n+1).

Sum_{k=0..n-1} T(n, k) = A058877(n+1), for n >= 1.

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A084633(n). (End)

Extensions

Missing term inserted by R. J. Mathar, Aug 06 2016