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

0,5

Links

Table of n, a(n) for n=0..54.

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

Example

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]

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

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.

Sequence in context: A155971 A176480 A154218 * A307139 A078121 A333157

Adjacent sequences: A326323 A326324 A326325 * A326327 A326328 A326329

Keyword

nonn,tabl

Author

Peter Luschny, Jul 02 2019

Status

approved