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A008779
Number of n-dimensional partitions of 5.
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
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
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
Sequence in context: A100454 A081436 A024205 * A062449 A014205 A002969
KEYWORD
nonn,easy
EXTENSIONS
More terms from Vincenzo Librandi, May 21 2015
STATUS
approved