A260917 Expansion of 1/(1 - x - x^2 - x^3 + x^6 + x^7).

1, 1, 2, 4, 7, 13, 23, 41, 74, 132, 236, 422, 754, 1348, 2409, 4305, 7694, 13750, 24573, 43915, 78481, 140255, 250652, 447944, 800528, 1430636, 2556712, 4569140, 8165581, 14592837, 26079086, 46606340, 83290915, 148850489, 266013023, 475396009, 849587598, 1518311204, 2713397556, 4849154954, 8666000202

Offset

0,3

Comments

This sequence counts the partially ordered partitions of (n) into parts 1,2,3,4 where the order (position) of adjacent pairs (1,3);(3,4);(2,4) is unimportant. Alternatively the order of complementary pairs (1,2);(1,4);(2,3) is important.

Links

Table of n, a(n) for n=0..40.

Index entries for related partition-counting sequences

Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,0,-1,-1).

Formula

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-6) - a(n-7).

G.f.: 1/((1 - x)*(1 - x^2 - 2*x^3 - 2*x^4 - 2*x^5 - x^6)).

Example

a(7)=41; the corresponding partitions (cf. comment) are: (43), (241=421), (124=142), (412), (214), (4111), (1411), (1141), (1114), (331=313=133), (322), (232), (223), (3112=1312=1132), (2113=2131=2311), (1213=1231), (3121=1321), (3211), (1123), (31111=13111=11311=11131=11113), (2221)=four, (22111)=ten, (211111)=six, (1111111).

Mathematica

CoefficientList[Series[1/(1 - x - x^2 - x^3 + x^6 + x^7), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2015 *)
LinearRecurrence[{1, 1, 1, 0, 0, -1, -1}, {1, 1, 2, 4, 7, 13, 23}, 50] (* Harvey P. Dale, Aug 21 2021 *)

Prog

(PARI) Vec(1/(1 - x - x^2 - x^3 + x^6 + x^7) + O(x^50)) \\ Michel Marcus, Aug 06 2015
(Magma) I:=[1, 1, 2, 4, 7, 13, 23]; [n le 7 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) - Self(n-6) - Self(n-7): n in [1..45]]; // Vincenzo Librandi, Aug 07 2015

Crossrefs

Cf. A023435, A080239, A260710.

Sequence in context: A003116 A368299 A303666 * A165648 A287381 A078038

Adjacent sequences: A260914 A260915 A260916 * A260918 A260919 A260920

Keyword

nonn,easy

Author

David Neil McGrath, Aug 04 2015

Status

approved