login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A374511
Expansion of 1/(1 - 4*x - 4*x^2)^(5/2).
3
1, 10, 80, 560, 3640, 22512, 134400, 781440, 4451040, 24939200, 137865728, 753625600, 4080643840, 21916106240, 116877312000, 619457482752, 3265293719040, 17128725519360, 89462514606080, 465434423336960, 2412895587536896, 12468681310412800, 64242981906022400
OFFSET
0,2
FORMULA
a(0) = 1, a(1) = 10; a(n) = (2*(2*n+3)*a(n-1) + 4*(n+3)*a(n-2))/n.
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = 2^(n-3)*Pochhammer(n+1, 4)*hypergeom([(1-n)/2, -n/2], [3], 2)/3. - Stefano Spezia, Jul 10 2024
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024
MATHEMATICA
a[n_]:=2^(n-3) Pochhammer[n+1, 4]*Hypergeometric2F1[(1-n)/2, -n/2, 3, 2]/3; Array[a, 23, 0] (* Stefano Spezia, Jul 10 2024 *)
PROG
(PARI) a(n) = binomial(n+4, 2)/6*sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jul 09 2024
STATUS
approved