A008778 a(n) = (n+1)*(n^2 +8*n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.

1, 5, 13, 26, 45, 71, 105, 148, 201, 265, 341, 430, 533, 651, 785, 936, 1105, 1293, 1501, 1730, 1981, 2255, 2553, 2876, 3225, 3601, 4005, 4438, 4901, 5395, 5921, 6480, 7073, 7701, 8365, 9066, 9805, 10583, 11401, 12260, 13161, 14105, 15093, 16126, 17205, 18331

Offset

0,2

Comments

Let m(i,1)=i; m(1,j)=j; m(i,j)=m(i-1,j)-m(i-1,j-1); then a(n)=m(n+3,3) - Benoit Cloitre, May 08 2002

a(n) = number of (n+6)-bit binary sequences with exactly 6 1's none of which is isolated. - David Callan, Jul 15 2004

If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-4) is the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007

Sum of first n triangular numbers plus previous triangular number. - Vladimir Joseph Stephan Orlovsky, Oct 13 2009

a(n) = Sum of first (n+1) triangular numbers plus n-th triangular number (see penultimate formula by Henry Bottomley). - Vladimir Joseph Stephan Orlovsky, Oct 13 2009

For n > 0, a(n-1) is the number of compositions of n+6 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016

The binomial transform of [1,4,4,1,0,0,0,...], the 4th row in A116672. - R. J. Mathar, Jul 18 2017

References

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190 eq. (11.4.7).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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

Francisco Javier de Vega, Some Variants of Integer Multiplication, Axioms (2023) Vol. 12, 905. See p. 15.

Milan Janjic, Two Enumerative Functions

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

László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = dot_product(n, n-1, ...2, 1)*(2, 3, ..., n, 1) for n = 2, 3, 4, ... [i.e., a(2) = (2, 1)*(2, 1), a(3) = (3, 2, 1)*(2, 3, 1)]. - Clark Kimberling

a(n) = a(n-1) + A034856(n+1) = A000297(n-1) + 1 = A000217(n) + A000292(n+1) = A000290(n-1) + A000292(n). - Henry Bottomley, Oct 25 2001

a(n) = Sum_{0<=k, l<=n; k+l|n} k*l. - Ralf Stephan, May 06 2005

G.f.: (1+x-x^2)/(1-x)^4. - Colin Barker, Jan 06 2012

a(n) = A000330(n+1) - A000292(n-1). - Bruce J. Nicholson, Jul 05 2018

E.g.f.: (6 +24*x +12*x^2 +x^3)*exp(x)/6. - G. C. Greubel, Sep 11 2019

Example

G.f. = 1 + 5*x + 13*x^2 + 26*x^3 + 45*x^4 + 71*x^5 + 105*x^6 + 148*x^7 + 201*x^8 + ...

Maple

seq(1+4*k+4*binomial(k, 2)+binomial(k, 3), k=0..45);

Mathematica

Table[(n+1)*(n^2+8*n+6)/6, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009, modified by G. C. Greubel, Sep 11 2019 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 5, 13, 26}, 51] (* G. C. Greubel, Sep 11 2019 *)

Prog

(Magma) [(n+1)*(n^2+8*n+6)/6: n in [0..50]]; // Vincenzo Librandi, May 21 2011
(PARI) Vec((1+x-x^2)/(1-x)^4 + O(x^50)) \\ Altug Alkan, Jan 07 2016
(Sage) [(n+1)*(n^2 +8*n +6)/6 for n in (0..50)] # G. C. Greubel, Sep 11 2019
(GAP) List([0..50], n-> (n+1)*(n^2 +8*n +6)/6); # G. C. Greubel, Sep 11 2019

Crossrefs

Cf. A022811-A022817, A024207-A024210.

Column 1 of triangle A094415.

Row n=4 of A022818.

Cf. A002411, A008779, A005712 (partial sums), A034856 (first diffs).

Sequence in context: A301300 A301677 A299259 * A299277 A014813 A180671

Adjacent sequences: A008775 A008776 A008777 * A008779 A008780 A008781

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved