A108489 Expansion of 1/sqrt(1-2x-5x^2-6x^3+9x^4).

1, 1, 4, 13, 37, 130, 427, 1441, 4954, 16987, 58843, 204610, 713893, 2500183, 8778478, 30898309, 108987427, 385136680, 1363252603, 4832572951, 17153677534, 60961916965, 216887253409, 772400234074, 2753261490919, 9822393082513

Offset: 0

Author

Paul Barry, Jun 04 2005

Keywords

• easy
• nonn

Comments

In general, Sum_{k=0..n}, C(n-k,k)^2*a^k*b^(n-k) has expansion 1/sqrt(1-2bx-(2ab-b^2)x^2-2a*b^2*x^3+(ab)^2*x^4).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1785
• Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Crossrefs

Cf. A108484.

Programs

Mathematica

Array[Sum[Binomial[# - k, k]^2*3^k, {k, 0, #}] &, 26, 0] (* Michael De Vlieger, Sep 10 2021 *)

Formula

a(n) = Sum_{k=0..n}, C(n-k, k)^2*3^k.

D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +5*(-n+1)*a(n-2) +3*(-2*n+3)*a(n-3) +9*(n-2)*a(n-4)=0. - R. J. Mathar, Feb 20 2015