A326326 T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for 0 <= k <= n, triangle read by rows.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 15, 7, 1, 1, 34, 65, 42, 11, 1, 1, 154, 339, 267, 96, 16, 1, 1, 874, 2103, 1891, 831, 191, 22, 1, 1, 5914, 15171, 15023, 7600, 2151, 344, 29, 1, 1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1

Offset: 0

Author

Peter Luschny, Jul 02 2019

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 2,     1]
[3] [1, 4,     4,      1]
[4] [1, 10,    15,     7,      1]
[5] [1, 34,    65,     42,     11,    1]
[6] [1, 154,   339,    267,    96,    16,    1]
[7] [1, 874,   2103,   1891,   831,   191,   22,   1]
[8] [1, 5914,  15171,  15023,  7600,  2151,  344,  29,  1]
[9] [1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1]
		

Crossrefs

Same construction for the falling factorial is A176663.

The inverse of the lower triangular matrix is the signed form of A256894.

Second column is A003422(n) and row sums are A003422(n+1).

Alternating row sums are A000007.

Third column is A097422.

Cf. A265609, A132393.

Programs

Maple

with(PolynomialTools):
T_row := n -> CoefficientList(expand(add(pochhammer(x, j), j=0..n)),x):
ListTools:-Flatten([seq(T_row(n), n=0..9)]);

Mathematica

Table[CoefficientList[FunctionExpand[Sum[Pochhammer[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten

Formula

Sum_{k=0..n} T(n, k)*x^k = Sum_{k=0..n} (x)^k, where (x)^k denotes the rising factorial.

Conjecture: T(n,k) = Sum_{i=0..n} A132393(i,k) for 0 <= k <= n. - Werner Schulte, Mar 30 2022