A200756 Triangle T(n,k) = coefficient of x^n in expansion of ((1 -sqrt(1 - 4*x - 4*x^2))/2)^k.

1, 2, 1, 4, 4, 1, 12, 12, 6, 1, 40, 40, 24, 8, 1, 144, 144, 92, 40, 10, 1, 544, 544, 360, 176, 60, 12, 1, 2128, 2128, 1440, 752, 300, 84, 14, 1, 8544, 8544, 5872, 3200, 1400, 472, 112, 16, 1, 35008, 35008, 24336, 13664, 6352, 2400, 700, 144, 18, 1

Offset: 1

Author

Vladimir Kruchinin, Nov 22 2011

Keywords

• nonn
• tabl

Comments

Triangle T(n,k) =

1. Riordan Array (1, (1 - sqrt(1 - 4*x - 4*x^2))/2) without first column.

2. Riordan Array ((1 - sqrt(1 - 4*x - 4*x^2))/(2*x), (1 - sqrt(1 - 4*x - 4*x^2))/2) numbering triangle (0,0).

The array factorizes in the Bell subgroup of the Riordan group as (1 + x, x*(1 + x)) * (c(x), x*c(x)) = A030528 * A033184, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. - Peter Bala, Dec 11 2015

Examples

			    1,
    2,   1,
    4,   4,   1,
   12,  12,   6,   1,
   40,  40,  24,   8,  1,
  144, 144,  92,  40, 10,  1,
  544, 544, 360, 176, 60, 12, 1
		

Crossrefs

Cf. A025227 (column 1), A000108, A030528, A033184.

Programs

Mathematica

Table[k Sum[Binomial[i, n - i] Binomial[-k + 2 i - 1, i - 1]/i, {i, k, n}], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Dec 11 2015 *)

Maxima

T(n,k):=k*(sum((binomial(i,n-i)*binomial(-k+2*i-1,i-1))/i,i,k,n));

PARI

tabl(nn) = {for (n=1, nn, for(k=1, n, print1(k*sum(i=k, n, binomial(i,n-i)*binomial(-k+2*i-1,i-1)/i),", ",);); print();); };
tabl(10); \\ Indranil Ghosh, Mar 04 2017

Formula

T(n,k) = k*( Sum_{i = k..n} binomial(i,n-i)*binomial(-k+2*i-1,i-1)/i );