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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			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 + ...
		

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).

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).

Programs

GAP

List([0..50], n-> (n+1)*(n^2 +8*n +6)/6); # G. C. Greubel, Sep 11 2019

Magma

[(n+1)*(n^2+8*n+6)/6: n in [0..50]]; // Vincenzo Librandi, May 21 2011

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 *)

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

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