A371771 a(n) = Sum_{k=0..floor(n/3)} binomial(4*n-3*k-1,n-3*k).

1, 3, 21, 166, 1377, 11748, 102088, 898677, 7987305, 71517307, 644134026, 5829345492, 52964836184, 482846377185, 4414405051413, 40458397722306, 371605426607673, 3419639400458316, 31521758873514301, 291000881055737811, 2690082750919841442

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = [x^n] 1/((1-x^3) * (1-x)^(3*n)).

a(n) = binomial(4*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-4*n)/3, 2*(1-2*n)/3, 1-4*n/3], 1). - Stefano Spezia, Apr 06 2024

From Vaclav Kotesovec, Apr 08 2024: (Start)

Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(9037*n^4 - 61391*n^3 + 154035*n^2 - 169317*n + 68836)*a(n) = 27*(2394805*n^7 - 19820156*n^6 + 66654684*n^5 - 117198990*n^4 + 115250735*n^3 - 62650734*n^2 + 17209736*n - 1814400)*a(n-1) - 3*(7021749*n^7 - 58192764*n^6 + 196050236*n^5 - 345531070*n^4 + 340849311*n^3 - 186035886*n^2 + 51353864*n - 5443200)*a(n-2) + 8*(2*n - 3)*(4*n - 9)*(4*n - 7)*(9037*n^4 - 25243*n^3 + 24084*n^2 - 9272*n + 1200)*a(n-3).

a(n) ~ 2^(8*n + 9/2) / (7 * sqrt(Pi*n) * 3^(3*n + 3/2)). (End)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(4*n-3*k-1, n-3*k));

Crossrefs

Cf. A005810, A147855, A262977.

Cf. A371758, A371770, A371772.

Sequence in context: A371817 A262977 A361375 * A214391 A370284 A046637

Adjacent sequences: A371768 A371769 A371770 * A371772 A371773 A371774

Keyword

nonn

Author

Seiichi Manyama, Apr 05 2024

Status

approved