A369809 Expansion of 1/(1 - x^6/(1-x)^7).

1, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 925, 1730, 3108, 5565, 10388, 20944, 45697, 104673, 242481, 553455, 1229305, 2650221, 5565127, 11465758, 23397041, 47757235, 98317135, 205108561, 433747259, 926655972, 1989584722, 4271185538, 9133958765, 19421679515

Offset: 0

Author

Seiichi Manyama, Feb 01 2024

Keywords

• nonn
• easy

Comments

Number of compositions of 7*n-6 into parts 6 and 7.

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-6,1).

Crossrefs

Cf. A099253, A369805, A369806, A369807, A369808.

Cf. A088305, A095263, A290998, A368475, A369794.

Cf. A000579, A017847.

Programs

PARI

my(N=40, x='x+O('x^N)); Vec(1/(1-x^6/(1-x)^7))

PARI

a(n) = sum(k=0, n\6, binomial(n-1+k, n-6*k));

Formula

G.f. (1-x)^7/((1-x)^7-x^6).

a(n) = A017847(7*n-6) = Sum_{k=0..floor((7*n-6)/6)} binomial(k,7*n-6-6*k) for n > 0.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7) for n > 7.

a(n) = Sum_{k=0..floor(n/6)} binomial(n-1+k,n-6*k).

a(n) = A373912(n)-A373912(n-1). - R. J. Mathar, Jun 24 2024