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URL: https://oeis.org/A326326

⇱ A326326 - OEIS

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A326326
T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for 0 <= k <= n, triangle read by rows.
2
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
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.
Sequence in context: A176480 A154218 A373744 * A307139 A078121 A333157
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jul 02 2019
STATUS
approved