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

0,5

Comments

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

Links

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

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.

Example

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;
  ...

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 *)

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.

Sequence in context: A206294 A058150 A058151 * A137651 A248826 A058152

Adjacent sequences: A257670 A257671 A257672 * A257674 A257675 A257676

Keyword

nonn,tabl

Author

Alois P. Heinz, May 03 2015

Status

approved