A257673 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: row n is the inverse binomial transform of the n-th row of array A255961, which has the Euler transform of (j->j*k) in column k.

1, 0, 1, 0, 3, 1, 0, 6, 6, 1, 0, 13, 21, 9, 1, 0, 24, 62, 45, 12, 1, 0, 48, 162, 174, 78, 15, 1, 0, 86, 396, 576, 376, 120, 18, 1, 0, 160, 917, 1719, 1509, 695, 171, 21, 1, 0, 282, 2036, 4761, 5340, 3285, 1158, 231, 24, 1, 0, 500, 4380, 12441, 17234, 13473, 6309, 1792, 300, 27, 1

Offset: 0

Author

Alois P. Heinz, May 03 2015

Keywords

• nonn
• tabl

Comments

T is the convolution triangle of the number of plane partitions (A000219). - Peter Luschny, Oct 19 2022

Examples

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,    1;
  0,   6,    6,    1;
  0,  13,   21,    9,    1;
  0,  24,   62,   45,   12,    1;
  0,  48,  162,  174,   78,   15,    1;
  0,  86,  396,  576,  376,  120,   18,   1;
  0, 160,  917, 1719, 1509,  695,  171,  21,  1;
  0, 282, 2036, 4761, 5340, 3285, 1158, 231, 24, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000007, A000219 (for n>0), A321947, A321948, A321949, A321950, A321951, A321952, A321953, A321954, A321955.

Main diagonal and lower diagonals give: A000012, A008585, A081266.

Row sums give A257674.

T(2n,n) give A257675.

Cf. A255961.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# Uses function PMatrix from A357368.
PMatrix(10, A000219); # Peter Luschny, Oct 19 2022

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A255961(n,k-i).

G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j)^j)^k.