A137597 - OEIS

%I #13 Aug 31 2023 10:54:36

%S 1,2,1,5,5,1,15,22,9,1,52,99,61,14,1,203,471,385,135,20,1,877,2386,

%T 2416,1140,260,27,1,4140,12867,15470,9156,2835,455,35,1,21147,73681,

%U 102215,72590,28441,6230,742,44,1

%N Triangle read by rows: A008277 * A007318.

%C Row sums = A035009 starting (1, 3, 11, 47, 227, ...).

%H Michael De Vlieger, <a href="/A137597/b137597.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)

%H Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, <a href="https://arxiv.org/abs/2308.14183">Combinatorial Identities for Vacillating Tableaux</a>, arXiv:2308.14183 [math.CO], 2023. See p. 16.

%F A008277 * A007318 as infinite lower triangular matrices.

%e First few rows of the triangle:

%e     1;

%e     2,   1;

%e     5,   5,   1;

%e    15,  22,   9,   1;

%e    52,  99,  61,  14,  1;

%e   203, 471, 385, 135, 20, 1;

%e   ...

%p T:= (n, k)-> add(Stirling2(n, j)*binomial(j-1, k-1), j=k..n):

%p seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Sep 03 2019

%t Table[Sum[StirlingS2[n, j]*Binomial[j - 1, k - 1], {j, k, n}], {n, 9}, {k, n}] // Flatten (* _Michael De Vlieger_, Aug 31 2023 *)

%Y Cf. A035009, A008277.

%Y Cf. A126350.

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Jan 29 2008