A390045 Expansion of 1 / ((1-x)^5 - x^6).

1, 5, 15, 35, 70, 126, 211, 340, 550, 935, 1716, 3367, 6826, 13835, 27490, 53176, 100283, 185573, 340006, 622575, 1147796, 2138884, 4028036, 7642004, 14542101, 27646776, 52379410, 98807200, 185669195, 348018300, 651763377, 1221316021, 2291969970, 4308708305

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1,1).

Formula

a(n) = Sum_{k=0..floor(n/6)} binomial(n-k+4,n-6*k).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + a(n-6).

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1, 1}, {1, 5, 15, 35, 70, 126}, 50] (* Vincenzo Librandi, Jan 16 2026 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(1/((1-x)^5-x^6))
(Magma) I:=[1, 5, 15, 35, 70, 126]; [n le 6 select I[n] else 5*Self(n-1) - 10*Self(n-2) + 10*Self(n-3) - 5*Self(n-4) + Self(n-5) + Self(n-6): n in [1..40]]; // Vincenzo Librandi, Jan 16 2026

Crossrefs

Cf. A049016, A390044.

Sequence in context: A000332 A392546 A342213 * A373936 A140227 A264925

Adjacent sequences: A390042 A390043 A390044 * A390046 A390047 A390048

Keyword

nonn,easy

Author

Seiichi Manyama, Jan 15 2026

Status

approved