OFFSET
0,3
COMMENTS
Row sums are A032443(n).
LINKS
Muniru A Asiru, Rows n=0..100 of triangle, flattened
FORMULA
T(n, m) = binomial(n,m)*A008949(n,m). [Nov 03 2009]
G.f.: (1/x)*d(arctanh(N(x,y)))/dy, where N(x,y) is g.f. of Narayana numbers (A001263). - Vladimir Kruchinin, Apr 11 2018
T(n, k) = binomial(n, k)*(2^n - binomial(n, 1+k)*hypergeom([1, 1+k-n], [k+2], -1)). - Peter Luschny, Dec 28 2018
From Natalia L. Skirrow, Nov 17 2025: (Start)
G.f.: f(x,y) = (1 - (x*(3+y)-1)/sqrt((1-x*(1+y))^2 - 4*x^2*y))/(2*(1-2*x*(1+y))).
G.f. f(x,y) satisfies f(x,y) - (1-2*x*(1+y))*f(x,y)^2 = x/((1-x*(1+y))^2 - 4*x^2*y).
D-finite with k*(n-k)*T(n,k) = n * ((n-k)*T(n-1,k-1) + k*T(n-1,k)).
D-finite with k^2*T(n,k) = 2*n^2*T(n-1,k-1) - (n-k+1)*(n-k)*T(n,k-1). (End)
EXAMPLE
n\k| 0 1 2 3 4 5 6 7 8 9 10
---+-------------------------------------------------------------
0 | 1,
1 | 1, 2,
2 | 1, 6, 4,
3 | 1, 12, 21, 8,
4 | 1, 20, 66, 60, 16,
5 | 1, 30, 160, 260, 155, 32,
6 | 1, 42, 330, 840, 855, 378, 64,
7 | 1, 56, 609, 2240, 3465, 2520, 889, 128,
8 | 1, 72,1036, 5208,11410, 12264, 6916, 2040, 256,
9 | 1, 90,1656,10920,32256, 48132, 39144, 18072, 4599, 512,
10 | 1,110,2520,21120,81060,160776,178080,116160,45585,10230,1024
MAPLE
T:=(n, m)-> binomial(n, m)*add(binomial(n, k), k=0..m): seq(seq(T(n, m), m=0..n), n=0..9); # Muniru A Asiru, Dec 28 2018
MATHEMATICA
T[m_, n_] = If[m == 0 && n == 0, 1, Sum[Binomial[m, n]*Binomial[m, k], {k, 0, n}]]
Flatten[Table[Table[T[m, n], {n, 0, m}], {m, 0, 10}]]
T[n_, k_] := Binomial[n, k] (2^n - Binomial[n, k + 1] Hypergeometric2F1[1, 1 -n + k, k + 2, -1]); Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 28 2018 *)
PROG
(GAP) t:=Flat(List([0..10], n->List([0..n], m->Binomial(n, m)*Sum([0..m], k->Binomial(n, k)))));; Print(t); # Muniru A Asiru, Dec 28 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Oct 27 2009
EXTENSIONS
Introduced OEIS notational standards in the definition - The Assoc. Editors of the OEIS, Nov 05 2009
STATUS
approved
