A200965 Triangle T(n,k) = coefficient of x^n in expansion of ((1-sqrt(1-4*x))/((1-x)*2))^k = sum(n>=k, T(n,k) * x^n).

1, 2, 1, 4, 4, 1, 9, 12, 6, 1, 23, 34, 24, 8, 1, 65, 98, 83, 40, 10, 1, 197, 294, 273, 164, 60, 12, 1, 626, 919, 891, 612, 285, 84, 14, 1, 2056, 2974, 2938, 2188, 1195, 454, 112, 16, 1, 6918, 9891, 9846, 7698, 4677, 2118, 679, 144, 18, 1, 23714, 33604, 33549

Offset

1,2

Comments

Triangle T(n,k)=

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

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

Convolution triangle of A014137(n). - Philippe Deléham, Jan 23 2014

Links

Table of n, a(n) for n=1..58.

Formula

T(n,k):=k*sum(i=0..n-k, (binomial(i+k-1,k-1)*binomial(2*(n-i)-k-1,n-i-1))/(n-i)).

Example

Triangle:
1,
2, 1,
4, 4, 1,
9, 12, 6, 1,
23, 34, 24, 8, 1,
65, 98, 83, 40, 10, 1,
197, 294, 273, 164, 60, 12, 1

Mathematica

T[n_, k_]:= (k/n) (Binomial[-1 - k + 2 n, -1 + n] HypergeometricPFQ[{k, k - n, -n}, {1/2 + k/2 - n, 1 + k/2 - n}, 1/4]);
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // TableForm (* Peter Luschny, May 30 2022 *)

Prog

(Maxima)
T(n, k):=k*sum((binomial(i+k-1, k-1)*binomial(2*(n-i)-k-1, n-i-1))/(n-i), i, 0, n-k);

Crossrefs

Cf. Columns: A014137, A014143

Sequence in context: A209240 A263989 A202710 * A117427 A097761 A200756

Adjacent sequences: A200962 A200963 A200964 * A200966 A200967 A200968

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Nov 25 2011

Status

approved