A377237 Expansion of 1/sqrt(1 - 4*x/sqrt(1 - 4*x)).

1, 2, 10, 56, 326, 1936, 11644, 70672, 431942, 2654816, 16392564, 101611536, 631938524, 3941350816, 24643020344, 154415141152, 969445760070, 6096812777664, 38401653547204, 242213348616592, 1529642560685684, 9671100898555168, 61208631472013256, 387759384222157152

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1200

Formula

a(n) = 4^n * Sum_{k=0..n} (-1)^k * binomial(-1/2,k) * binomial(n-k/2-1,n-k).

a(n) ~ 2^(n+1) * (1 + sqrt(5))^(n - 1/2) / (5^(1/4) * sqrt(Pi*n)). - Vaclav Kotesovec, May 03 2025

D-finite with recurrence: 32*(4*n + 3)*(4*n + 1)*a(n) - 16*(24*n^2 + 100*n + 85)*a(n + 1) + 8*(97 + 44*n)*a(n + 2) + 4*(14*n^2 + 67*n + 72)*a(n + 3) - 2*(n + 4)*(7*n + 23)*a(n + 4) + (n + 5)*(n + 4)*a(n + 5) = 0. - Robert Israel, Feb 24 2026

Maple

f:= rectoproc({32*(4*n + 3)*(4*n + 1)*a(n) - 16*(24*n^2 + 100*n + 85)*a(n + 1) + 8*(97 + 44*n)*a(n + 2) + 4*(14*n^2 + 67*n + 72)*a(n + 3) - 2*(n + 4)*(7*n + 23)*a(n + 4) + (n + 5)*(n + 4)*a(n + 5), a(0) = 1, a(1) = 2, a(2) = 10, a(3) = 56, a(4) = 326} , a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 24 2026

Prog

(PARI) a(n) = 4^n*sum(k=0, n, (-1)^k*binomial(-1/2, k)*binomial(n-k/2-1, n-k));

Crossrefs

Cf. A011782, A377238.

Cf. A377239.

Sequence in context: A108490 A323935 A370195 * A165817 A243644 A000172

Adjacent sequences: A377234 A377235 A377236 * A377238 A377239 A377240

Keyword

nonn

Author

Seiichi Manyama, Oct 21 2024

Status

approved