A125027 Binomial transform of the "1,2,3,..." triangle.

1, 3, 3, 9, 11, 6, 26, 32, 27, 10, 72, 86, 85, 54, 15, 192, 222, 233, 189, 95, 21, 496, 558, 597, 549, 371, 153, 28, 1248, 1374, 1473, 1446, 1160, 664, 231, 36, 3072, 3326, 3549, 3600, 3203, 2246, 1107, 332, 45, 7424, 7934, 8409, 8659, 8201, 6567, 4051, 1745, 459, 55, 17664, 18686, 19669, 20367, 20015, 17503, 12597, 6893, 2629, 615, 66

Offset: 1

Author

Gary W. Adamson, Nov 15 2006

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle:
   1;
   3,  3;
   9, 11,  6;
  26, 32, 27, 10;
  72, 86, 85, 54, 15;
  ...
		

Crossrefs

Cf. A072863 (first column), A000217 (diagonal), A164845 (subdiagonal).

Programs

Maple

A27 := proc(n,k)
    option remember;
    if k>= 0 and k <=n then
        if k = 0 then
            1+procname(n-1,n-1) ;
        else
            procname(n,0)+k ;
        end if;
    else
        0;
    end if;
end proc:
A125027 := proc(n,k)
    add( binomial(n,j)*A27(j,k),j=k..n) ;
end proc: # R. J. Mathar, May 21 2018

Mathematica

A27[n_, k_] := A27[n, k] = If[k >= 0 && k <= n, If[k == 0, 1+A27[n-1, n-1], A27[n, 0]+k], 0];
A125027[n_, k_] := Sum[Binomial[n, j]*A27[j, k], {j, k, n}];
Table[A125027[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2024, after R. J. Mathar *)

Formula

Given the triangle (natural numbers in succession: 1; 2,3; 4,5,6; ...) as an infinite matrix M and P = Pascal's triangle as a lower triangular matrix, perform P*M, deleting the zeros.

The row sums s(n) = 1, 6, 26, 95, 312, 952, ... obey (-3*n+2)*s(n) +(9*n+7)*s(n-1) + 2*(-3*n-2)*s(n-2) = 0. - R. J. Mathar, May 21 2018