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

1,2

Links

Table of n, a(n) for n=1..66.

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A121072 A133164 A022156 * A360244 A360242 A005296

Adjacent sequences: A125024 A125025 A125026 * A125028 A125029 A125030

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Nov 15 2006

Status

approved