OFFSET
0,8
COMMENTS
Conjecture: Generally, for k > 0 is a(n) ~ n^k * ((1+sqrt(5))/2)^(n-2*k-1) / (5^((k+1)/2) * k!). Holds for all k<=10. - Vaclav Kotesovec, May 02 2014
G.f.: 1 + Sum_{i>0} (-y*(x^i)*(x - 1)^2)/( (x^(i+1) + x - 1)*((x^i)*(x*(y - 1) - y) - x + 1) ). - John Tyler Rascoe, Oct 15 2024
Sum_{k=0..n} k * T(n,k) = A097941(n). - Alois P. Heinz, Oct 15 2024
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened
EXAMPLE
Triangle starts:
00: 1;
01: 0, 1;
02: 0, 1, 1;
03: 0, 3, 0, 1;
04: 0, 3, 4, 0, 1;
05: 0, 8, 3, 4, 0, 1;
06: 0, 11, 10, 5, 5, 0, 1;
07: 0, 20, 18, 14, 5, 6, 0, 1;
08: 0, 34, 35, 24, 21, 6, 7, 0, 1;
09: 0, 59, 60, 59, 35, 27, 7, 8, 0, 1;
10: 0, 96, 121, 108, 85, 49, 35, 8, 9, 0, 1;
11: 0, 167, 217, 213, 175, 125, 63, 44, 9, 10, 0, 1;
12: 0, 282, 391, 419, 366, 258, 176, 80, 54, 10, 11, 0, 1;
13: 0, 475, 709, 808, 730, 579, 371, 236, 99, 65, 11, 12, 0, 1;
14: 0, 800, 1281, 1522, 1481, 1202, 861, 513, 309, 120, 77, 12, 13, 0, 1;
15: 0, 1352, 2283, 2872, 2925, 2512, 1862, 1238, 684, 395, 143, 90, 13, 14, 0, 1;
...
MAPLE
b:= proc(n, s) option remember; `if`(n=0, 1,
`if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
end:
T:= (n, k)->`if`(k=0, `if`(n=0, 1, 0), add((p->add(coeff(p, x, i)*
binomial(i+k, k), i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)):
seq(seq(T(n, k), k=0..n), n=0..15);
MATHEMATICA
b[n_, s_] := b[n, s] = If[n == 0, 1, If[n<s, 0, Expand[Sum[b[n-j, s]*x, {j, s, n}]]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i+k, k], {i, 0, Exponent[p, x]}]][b[n-j*k, j+1]], {j, 1, n/k}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Nov 07 2014, translated from Maple *)
PROG
(PARI)
T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h=1+sum(i=1, N, (-y*(x^i)*(x-1)^2)/((x^(i+1)+x-1)*((x^i)*(x*(y-1)-y)-x+1)))); for(i=0, N-1, print(Vecrev(polcoef(h, i))))}
T_xy(15) \\ John Tyler Rascoe, Oct 15 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 25 2014
STATUS
approved