A391105 a(n) = binomial(4*n,n) - binomial(3*n,n).

0, 1, 13, 136, 1325, 12501, 116032, 1067760, 9782829, 89456455, 817615513, 7475802412, 68416856768, 626891134156, 5751871734720, 52849221767136, 486272088165933, 4480380512338923, 41335163417004835, 381823967694522700, 3531124297706368825, 32691615281688295845

Offset

0,3

Comments

It appears that a(n) mod n^3 == 1 iff n is an odd prime (tested to 500000).

In general, it appears that C((k+1)*n,n)-C(k*n,n) == 1 (mod n^k) if n is prime.

For k = 1, there are 60 pseudoprimes < 50000 with only two of them, 418 = 2*11*19 and 27173 = 29*937 not being a power of a prime. For k = 2, 49 is the only pseudoprime < 50000. For k = 3, 2 is the only pseudoprime < 50000.

Links

Robert Israel, Table of n, a(n) for n = 0..1020

Formula

a(n) = (1/n!)*((4*n)!/(3*n)! - (3*n)!/(2*n)!).

a(n) = Sum_{k=1..n} binomial(n,k)*binomial(3*n,n-k).

a(n) = (Product_{j=1..n} (3*n+j) - Product_{j=1..n} (2*n+j))/n!.

G.f.: hypergeom([1/4, 1/2, 3/4], [1/3, 2/3], 2^8*x/3^3) - 2*cos(arccos(1-27*x/2)/6)/sqrt(4 - 27*x). - Stefano Spezia, Nov 30 2025

D-finite with recurrence: 24*(4*n + 3)*(3*n + 2)*(2*n + 1)*(3*n + 1)*(4*n + 1)*(295*n^4 + 1770*n^3 + 3959*n^2 + 3912*n + 1440)*a(n) - (n + 1)*(517135*n^8 + 4137080*n^7 + 13890622*n^6 + 25424612*n^5 + 27575647*n^4 + 18030668*n^3 + 6893580*n^2 + 1395936*n + 112320)*a(n + 1) + 6*(295*n^4 + 590*n^3 + 419*n^2 + 124*n + 12)*(n + 1)*(2*n + 3)*(3*n + 4)*(3*n + 5)*(n + 2)*a(n + 2) = 0. - Robert Israel, May 20 2026

Maple

seq(binomial(4*n, n) - binomial(3*n, n), n=0..20);

Crossrefs

Cf. A030662, A006480.

Sequence in context: A019519 A243216 A112225 * A069511 A052262 A065550

Adjacent sequences: A391102 A391103 A391104 * A391106 A391107 A391108

Keyword

nonn,easy

Author

Gary Detlefs, Nov 29 2025

Status

approved