|
| |
|
|
A297705
|
|
a(n) = Sum_{k=0..n} binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -4).
|
|
3
|
|
|
|
1, 6, 56, 636, 8036, 108516, 1533316, 22389396, 335177396, 5116746276, 79350018276, 1246583463156, 19797057247956, 317304181980036, 5126097354722436, 83384214787592916, 1364582474360361716, 22450780862515853796, 371129420131691349796, 6161232472210183370676
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: (1 - 3*x - sqrt(1 - 18*x + x^2))/(6*x + 4*x^2).
(-2*n-4)*a(n+1) + (33*n+120)*a(n+2) + (52*n+179)*a(n+3) + (-3*n-15)*a(n+4) = 0.
(End) [Conjectures verified with Maple's FormalPowerSeries Module. - Peter Luschny, Nov 10 2022]
a(n) ~ 5^(1/4) * phi^(6*n - 3) / (sqrt(2*Pi) * (349 - 156*sqrt(5)) * n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 05 2018
O.g.f. A(x) = (1/x) * series reversion of x*(1 - 3*x)/((1 + x)*(1 + 2*x)). Cf. A114710. - Peter Bala, Nov 08 2022
a(n) = Sum_{k=0..n} (Sum_{j=k..n} 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)*binomial(2*j - k, j - k)/(j + 1)). - Detlef Meya, Jan 15 2024
|
|
|
MAPLE
|
f:= n -> simplify(add(binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), k=0..n)):
a := proc(n) option remember; if n < 4 then return [1, 6, 56, 636][n + 1] fi;
((-26*n^2 + 130*n - 156)*a(n - 4) + (423*n^2 - 1431*n + 1170)*a(n - 3) + (775*n^2 - 2441*n + 2106)*a(n - 2) + n*9*(13*n - 1)*a(n - 1))/(9*(n + 1)*n) end:
|
|
|
MATHEMATICA
|
a[n_] := Sum[Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4], {k, 0, n}]; Table[a[n], {n, 0, 17}]
a[n_] := Sum[Sum[4^(j - k)*(k + 1)*Binomial[n + j - k, 2*j - k]*Binomial[2*j - k, j - k]/(j + 1), {j, k, n}], {k, 0, n}]; Flatten[Table[a[n], {n, 0, 19}]] (* Detlef Meya, Jan 15 2024 *)
|
|
|
PROG
|
(PARI) a(n) = sum(k=0, n, sum(j=k, n, 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)*binomial(2*j - k, j - k)/(j + 1))) \\ Andrew Howroyd, Jan 15 2024
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
