0,5

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with 0 <= k <= n. T(n,k) = 0 for k > n.

Alois P. Heinz, Rows n = 0..19, flattened

T(n,k) = A324428(n,k) - A324428(n,k+1) for k > 0, T(n,0) = A000007(n).

Triangle T(n,k) begins:

1;

0, 1;

0, 2, 1;

0, 11, 3, 1;

0, 74, 24, 6, 1;

0, 652, 225, 57, 10, 1;

0, 7069, 2489, 678, 141, 17, 1;

0, 90946, 32326, 9375, 2107, 352, 28, 1;

0, 1353554, 483968, 146334, 35568, 6722, 832, 46, 1;

...

b:= proc(n, f, m, l, j) option remember; (k-> `if`(n<add(i, i=f)+m+

add(i, i=l), 0, `if`(n=0, 1, add(`if`(f[i]=0, 0, b(n-1,

subsop(i=0, f), m+l[1], [subsop(1=[][], l)[], 0], max(0, j-1))),

i=max(1, j+1)..min(k, n-1))+`if`(m=0, 0, m*b(n-1, f, m-1+l[1],

[subsop(1=[][], l)[], 0], max(0, j-1)))+b(n-1, f, m+l[1],

[subsop(1=[][], l)[], 1], max(0, j-1)))))(nops(l))

end:

A:= (n, k)-> `if`(n=0 or k<2, doublefactorial(2*n-1),

b(2*n-k+1, [1$k-1], 0, [0$k-1], k-1)):

T:= (n, k)-> `if`(n=k, 1, A(n, k)-A(n, k+1)):

seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, f_List, m_, l_List, j_] := b[n, f, m, l, j] = Function[k, If[n < Total[f] + m + Total[l], 0, If[n == 0, 1, Sum[If[f[[i]] == 0, 0, b[n - 1, ReplacePart[f, i -> 0], m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]], {i, Max[1, j + 1], Min[k, n - 1]}] + If[m == 0, 0, m*b[n - 1, f, m - 1 + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j-1]]] + b[n-1, f, m + l[[1]], Append[ReplacePart[ l, 1 -> Nothing], 1], Max[0, j - 1]]]]][Length[l]];

A[n_, k_] := If[n == 0 || k < 2, 2^(n-1) Pochhammer[3/2, n-1], b[2n-k+1, Table[1, {k - 1}], 0, Table[0, {k - 1}], k - 1]];

T[n_, k_] := If[n == k, 1, A[n, k] - A[n, k + 1]];

Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

Columns k=0-10 give: A000007, A324445, A324446, A324447, A324448, A324449, A324450, A324451, A324452, A324453, A324454.

Row sums give A001147.

Main diagonal gives A000012.

T(n+1,n) gives A001610.

Cf. A293157, A293881, A324428.

Sequence in context: A185285 A268434 A010107 * A119830 A268435 A392857

Adjacent sequences: A324426 A324427 A324428 * A324430 A324431 A324432

nonn,tabl

Alois P. Heinz, Feb 27 2019

approved