A181546 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^4.

1, 1, 2, 17, 83, 338, 1923, 11553, 63028, 359203, 2172469, 13026034, 78106885, 478415635, 2957675956, 18321372721, 114301292581, 718253640196, 4531427831111, 28699590926291, 182566373639352, 1165539703613397

Offset

0,3

Comments

Conjecture: Given F(n,L) = Sum_{k=0..[n/2]} C(n-k,k)^L, then lim_{n->oo} F(n+1,L)/F(n,L) = (Fibonacci(L)*sqrt(5) + Lucas(L))/2 for L>=0 where Fibonacci(n) = A000045(n) and Lucas(n) = A000032(n).

For this sequence (L=4): lim_{n->oo} a(n+1)/a(n) = (3*sqrt(5)+7)/2 = 6.8541...

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y*z*w)^2). - Ilya Gutkovskiy, Apr 23 2025

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1202

C. Banderier, P. Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. Table 1.

Example

G.f. A(x) = 1 + x + 2*x^2 + 17*x^3 + 83*x^4 + 338*x^5 + 1923*x^6 +...
The terms begin:
a(0) = a(1) = 1^4;
a(2) = 1^4 + 1^4 = 2;
a(3) = 1^4 + 2^4 = 17;
a(4) = 1^4 + 3^4 + 1^4 = 83;
a(5) = 1^4 + 4^4 + 3^4 = 338;
a(6) = 1^4 + 5^4 + 6^4 + 1^4 = 1923;
a(7) = 1^4 + 6^4 + 10^4 + 4^4 = 11553; ...

Mathematica

Table[Sum[Binomial[n-k, k]^4, {k, 0, Floor[n/2]}], {n, 0, 30}] (* Harvey P. Dale, May 22 2021 *)

Prog

(PARI) {a(n)=sum(k=0, n\2, binomial(n-k, k)^4)}

Crossrefs

Cf. variants: A181545, A181547, A051286.

Cf. A000032, A000045.

Sequence in context: A309029 A079889 A053786 * A320644 A081744 A372189

Adjacent sequences: A181543 A181544 A181545 * A181547 A181548 A181549

Keyword

nonn

Author

Paul D. Hanna, Oct 29 2010

Status

approved