A132303 - OEIS

A132303

Sum of cubes of trinomial coefficients: a(n) = Sum_{k=0..2n} trinomial(n,k)^3 where trinomial(n,k) = [x^k] (1 + x + x^2)^n.

1, 3, 45, 831, 17181, 375903, 8530929, 198643455, 4714491357, 113550338127, 2767105469745, 68077260387315, 1688160321677025, 42142679453321307, 1058050429855640217, 26695057057648257231, 676431705046728704733, 17205315843416998571415, 439098128408223839364561, 11239967518370464873317291

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OFFSET

0,2

COMMENTS

Conjecture: the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) holds for all primes p >= 5 and positive integers n and k. - Peter Bala, Aug 29 2025

LINKS

Robert Israel, Table of n, a(n) for n = 0..698

MAPLE

f:= proc(n) local t; add(subs(x=1, t)^3, t = expand((1+x+x^2)^n)) end proc:

map(f, [$0..20]); # Robert Israel, Aug 29 2025

PROG

(PARI) a(n)=sum(k=0, 2*n, polcoeff((1+x+x^2)^n, k)^3)

CROSSREFS

Cf. A027907, A082758, A132304, A132305, A168593, A251686.

Sequence in context: A287031 A124487 A266698 * A386876 A298799 A202437

Adjacent sequences: A132300 A132301 A132302 * A132304 A132305 A132306

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 18 2007

STATUS

approved