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

1,2

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

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

    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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A200965 A117427 A097761 * A108556 A122440 A046943

Adjacent sequences: A200753 A200754 A200755 * A200757 A200758 A200759

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Nov 22 2011

Status

approved