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%I #11 Apr 11 2022 16:18:41
%S 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 1,1,874,2103,1891,831,191,22,1,1,5914,15171,15023,7600,2151,344,29,1,
%U 1,46234,124755,133147,74884,24600,4880,575,37,1
%N T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for 0 <= k <= n, triangle read by rows.
%F Sum_{k=0..n) T(n, k)*x^k = Sum_{k=0..n) (x)^k, where (x)^k denotes the rising factorial.
%F Conjecture: T(n,k) = Sum_{i=0..n} A132393(i,k) for 0 <= k <= n. - _Werner Schulte_, Mar 30 2022
%e Triangle starts:
%e [0] [1]
%e [1] [1, 1]
%e [2] [1, 2, 1]
%e [3] [1, 4, 4, 1]
%e [4] [1, 10, 15, 7, 1]
%e [5] [1, 34, 65, 42, 11, 1]
%e [6] [1, 154, 339, 267, 96, 16, 1]
%e [7] [1, 874, 2103, 1891, 831, 191, 22, 1]
%e [8] [1, 5914, 15171, 15023, 7600, 2151, 344, 29, 1]
%e [9] [1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1]
%p with(PolynomialTools):
%p T_row := n -> CoefficientList(expand(add(pochhammer(x, j), j=0..n)),x):
%p ListTools:-Flatten([seq(T_row(n), n=0..9)]);
%t Table[CoefficientList[FunctionExpand[Sum[Pochhammer[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten
%Y Same construction for the falling factorial is A176663.
%Y The inverse of the lower triangular matrix is the signed form of A256894.
%Y Second column is A003422(n) and row sums are A003422(n+1).
%Y Alternating row sums are A000007.
%Y Third column is A097422.
%Y Cf. A265609, A132393.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Jul 02 2019
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