A008779 Number of n-dimensional partitions of 5.

1, 7, 24, 59, 120, 216, 357, 554, 819, 1165, 1606, 2157, 2834, 3654, 4635, 5796, 7157, 8739, 10564, 12655, 15036, 17732, 20769, 24174, 27975, 32201, 36882, 42049, 47734, 53970, 60791, 68232, 76329, 85119, 94640, 104931, 116032, 127984, 140829, 154610, 169371

Offset

0,2

Comments

a(n) = number of (n+8)-bit binary sequences with exactly 8 1's none of which is isolated. - David Callan, Jul 15 2004

For n > 0, a(n) is the number of compositions of n+8 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016

Binomial transform of [1,6,11,7,1,0,0,0,...], the 5th row of A116672. - R. J. Mathar, Jul 18 2017

References

G. E. Andrews, The Theory of Partitions, Add.-Wes. '76, p. 190.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

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

Formula

G.f.: (1 +2*x -x^2 -x^3)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009

a(n) = (n+1)*(n^3 + 21*n^2 + 38*n + 24)/24. - M. F. Hasler, Sep 15 2009

a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5). - Vincenzo Librandi, May 21 2015

E.g.f.: (24 + 144*x + 132*x^2 + 28*x^3 + x^4)*exp(x)/24. - G. C. Greubel, Sep 11 2019

Maple

seq(1+6*n+11*binomial(n, 2)+7*binomial(n, 3)+binomial(n, 4), n=0..45);

Mathematica

CoefficientList[Series[(1+2*x-x^2-x^3)/(1-x)^5, {x, 0, 45}], x] (* Vincenzo Librandi, May 21 2015 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 7, 24, 59, 120}, 46] (* G. C. Greubel, Sep 11 2019 *)

Prog

(Magma) [(n+1)*(n^3+21*n^2+38*n+24)/24: n in [0..45]] /* or */ I:=[1, 7, 24, 59, 120]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..45]]; // Vincenzo Librandi, May 21 2015
(PARI) Vec((-1+x^3+x^2-2*x)/(x-1)^5 + O(x^45)) \\ Altug Alkan, Jan 07 2016
(Sage) [(n+1)*(n^3 + 21*n^2 + 38*n + 24)/24 for n in (0..45)] # G. C. Greubel, Sep 11 2019
(GAP) List([0..45], n-> (n+1)*(n^3 + 21*n^2 + 38*n + 24)/24); # G. C. Greubel, Sep 11 2019

Crossrefs

Cf. A116672, A289656.

Sequence in context: A100454 A081436 A024205 * A062449 A014205 A002969

Adjacent sequences: A008776 A008777 A008778 * A008780 A008781 A008782

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vincenzo Librandi, May 21 2015

Status

approved