A289656 -id:A289656 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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).

Cf. A116672, A289656.

List([0..45], n-> (n+1)*(n^3 + 21*n^2 + 38*n + 24)/24); # G. C. Greubel, Sep 11 2019

[(n+1)*(n^3+21*n^2+38*n+24)/24: n in [0..45]]; // Vincenzo Librandi, May 21 2015

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

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

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 *)

Vec((-1+x^3+x^2-2*x)/(x-1)^5 + O(x^45)) \\ Altug Alkan, Jan 07 2016

[(n+1)*(n^3 + 21*n^2 + 38*n + 24)/24 for n in (0..45)] # G. C. Greubel, Sep 11 2019

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

More terms from Vincenzo Librandi, May 21 2015

A116672 Triangle read by rows in which the binomial transform of the n-th row gives the Euler transform of the n-th diagonal of Pascal's triangle (A007318).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 7, 1, 1, 10, 27, 29, 12, 1, 1, 14, 57, 96, 72, 21, 1, 1, 21, 117, 277, 319, 176, 38, 1

Offset: 1

Alford Arnold, Feb 22 2006

For example, the Euler transform of 1,3,6,... is 1,1,4,10,26,59,141,... (A000294) differing slightly from A000293 which counts the solid partitions.
The NAME does not reproduce the DATA, COMMENTS, or EXAMPLES.  - R. J. Mathar, Jul 19 2017
The binomial transforms of the rows form the rows of A289656. - N. J. A. Sloane, Jul 19 2017

			Row 6 is 1 10 27 29 12 1 generating 1 11 48 141 ... (A008780) the seventh term in the Euler transforms of 1,1,1,...; 1,2,3,...; 1,3,6,... 1,4,10,... etc.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 6, 11, 7, 1;
1, 10, 27, 29, 12, 1;
1, 14, 57, 96, 72, 21, 1;
1, 21, 117, 277, 319, 176, 38, 1;
...

N. J. A. Sloane, Transforms

Cf. A000293, A116673 (row sums), A008778 - A008780, A289656.

cp's OEIS Frontend

A008779 Number of n-dimensional partitions of 5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions