A107025 Binomial transform of the expansion of 1/(1-x^5-x^6).

1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 938, 1808, 3459, 6826, 14198, 30960, 69143, 154433, 340006, 734561, 1561313, 3286129, 6900097, 14542101, 30855957, 65908862, 141395972, 303745077, 651763377, 1395140215, 2978858672

Offset: 0

Author

Paul Barry, May 09 2005

Keywords

• easy
• nonn

Comments

In general, the binomial transform of 1/(1-x^r-x^(r+1)) is given by (1-x)^r/((1-x)^(r+1)-x^r), with a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+k,(r+1)k) = Sum_{k=0..floor((r+1)n/r)} binomial(k,(r+1)n-r*k).

Number of compositions of 6*n into parts 5 and 6. - Seiichi Manyama, Jun 22 2024

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Crossrefs

Cf. A017837, A099099, A099131.

Formula

G.f.: (1-x)^5/((1-x)^6-x^5).

a(n) = 6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+7a(n-5)-a(n-6).

a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+k, 6k).

a(n) = Sum_{k=0..floor(6n/5)} binomial(k, 6n-5k).

a(n) = A017837(6*n). - Seiichi Manyama, Jun 22 2024