OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..400
FORMULA
a(n) = n! / (-2)^n * Sum_{k=0..n} (-4)^k * binomial(k - 1/4, k) / (n-k)!.
E.g.f.: exp(-x/2) * (1-2*x)^(-3/4).
E.g.f. A(x) satisifes 0 = A(x)*(x+1) + A'(x)*(2*x-1).
a(n) ~ (n/e)^(n+1/4) * 2^n * Gamma(1/4) / sqrt(Pi).
0 = +a(n)*(+a(n+1) + 3*a(n+2) - a(n+3)) + a(n+1)*(+a(n+1) + 5*a(n+2) - 2*a(n+3)) + a(n+2)*(+2*a(n+2)) for all integer n>=0.
EXAMPLE
G.f. = 1 + x + 4*x^2 + 22*x^3 + 166*x^4 + 1582*x^5 + 18232*x^6 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {3/4, -n}, {}, 4] / (-2)^n];
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[-x/2] (1 - 2 x)^(-3/4), {x, 0, n}]];
a[ n_] := If[ n < 0, 0, n! / (-2)^n Sum[ (-4)^k Binomial[ k - 1/4, k] / (n - k)!, {k, 0, n}]];
a[ n_] := If[ n < 0, 0, RecurrenceTable[ {a[k] == (2 k - 1) a[k + 1] + (k - 1) a[k], a[0] == a[1] == 1}, a, {k, n, n}]];
nxt[{n_, a_, b_}]:={n+1, b, (2n+1)b+a n}; NestList[nxt, {1, 1, 1}, 20][[All, 2]] (* Harvey P. Dale, Jan 16 2023 *)
PROG
(PARI) {a(n) = if( n<0, 0, n! / (-2)^n * sum(k=0, n, (-4)^k * binomial(k - 1/4, k) / (n-k)!))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( exp(-x / 2 + A) * (1 - 2*x + A)^(-3/4), n))};
(Magma) I:=[1, 4]; [1] cat [n le 2 select I[n] else (2*n-1)*Self(n-1) + (n-1)*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 03 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 28 2014
STATUS
approved