A390589 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k+3,n-k).

1, 6, 46, 376, 3181, 27492, 241088, 2136704, 19090180, 171642566, 1551167806, 14077358512, 128208471574, 1171160277832, 10726023380836, 98455061321728, 905513833456684, 8342789125417080, 76984637089505752, 711383432273465496, 6581893062809397533

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

G.f.: g^3/((1-4*x*g^3) * (1+x*g^3)) where g = 1+x*g^4 is the g.f. of A002293.

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

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n+4,k) * binomial(4*n-k+3,n-k).

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

Mathematica

Table[Sum[(-1)^k*Binomial[4*n-k+3, n-k], {k, 0, n}], {n, 0, 22}] (* Vincenzo Librandi, Nov 12 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(4*n-k+3, n-k));
(Magma) [&+[(-1)^k*Binomial(4*n-k+3, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Nov 12 2025

Crossrefs

Cf. A147855, A386821, A390547, A390592.

Cf. A002293, A390413.

Sequence in context: A301421 A383610 A288689 * A386811 A271933 A380685

Adjacent sequences: A390586 A390587 A390588 * A390590 A390591 A390592

Keyword

nonn

Author

Seiichi Manyama, Nov 11 2025

Status

approved