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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.

%I A326326 #15 Dec 15 2024 17:03:37
%S A326326 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,
%T A326326 1,1,874,2103,1891,831,191,22,1,1,5914,15171,15023,7600,2151,344,29,1,
%U A326326 1,46234,124755,133147,74884,24600,4880,575,37,1
%N A326326 T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for 0 <= k <= n, triangle read by rows.
%F A326326 Sum_{k=0..n} T(n, k)*x^k = Sum_{k=0..n} (x)^k, where (x)^k denotes the rising factorial.
%F A326326 Conjecture: T(n,k) = Sum_{i=0..n} A132393(i,k) for 0 <= k <= n. - _Werner Schulte_, Mar 30 2022
%e A326326 Triangle starts:
%e A326326 [0] [1]
%e A326326 [1] [1, 1]
%e A326326 [2] [1, 2,     1]
%e A326326 [3] [1, 4,     4,      1]
%e A326326 [4] [1, 10,    15,     7,      1]
%e A326326 [5] [1, 34,    65,     42,     11,    1]
%e A326326 [6] [1, 154,   339,    267,    96,    16,    1]
%e A326326 [7] [1, 874,   2103,   1891,   831,   191,   22,   1]
%e A326326 [8] [1, 5914,  15171,  15023,  7600,  2151,  344,  29,  1]
%e A326326 [9] [1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1]
%p A326326 with(PolynomialTools):
%p A326326 T_row := n -> CoefficientList(expand(add(pochhammer(x, j), j=0..n)),x):
%p A326326 ListTools:-Flatten([seq(T_row(n), n=0..9)]);
%t A326326 Table[CoefficientList[FunctionExpand[Sum[Pochhammer[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten
%Y A326326 Same construction for the falling factorial is A176663.
%Y A326326 The inverse of the lower triangular matrix is the signed form of A256894.
%Y A326326 Second column is A003422(n) and row sums are A003422(n+1).
%Y A326326 Alternating row sums are A000007.
%Y A326326 Third column is A097422.
%Y A326326 Cf. A265609, A132393.
%K A326326 nonn,tabl
%O A326326 0,5
%A A326326 _Peter Luschny_, Jul 02 2019