A026811 Number of partitions of n in which the greatest part is 5.

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 119, 141, 164, 192, 221, 255, 291, 333, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 1342, 1469, 1602, 1747, 1898, 2062, 2233, 2418, 2611, 2818, 3034, 3266, 3507, 3765

Offset

0,8

Comments

Essentially same as A001401: five zeros followed by A001401.

Also number of partitions of n into exactly 5 parts.

References

D. E. Knuth, The Art of Computer Programming, vol. 4, fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Links

Washington Bomfim, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

Formula

a(n) = round( ((n^4+10*(n^3+n^2)-75*n -45*n*(-1)^n)) / 2880 ). - Washington Bomfim, Jul 03 2012

G.f.: x^5/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)). - Joerg Arndt, Jul 04 2012

a(n) = A008284(n,5). - Robert A. Russell, May 13 2018

From Gregory L. Simay, Jul 28 2019: (Start)

a(2n) = a(2n-1) + a(n+1) + a(n) - a(n-3) - a(n-4);

a(2n+1) = a(2n) + a(n+3) - a(n-5). (End)

From R. J. Mathar, Jun 23 2021: (Start)

a(n) - a(n-5) = A001400(n-5).

a(n) - a(n-4) = A008669(n-5).

a(n) - a(n-3) = A029007(n-5).

a(n) - a(n-2) = A029032(n-5).

a(n) = +a(n-1) +a(n-2) -a(n-5) -a(n-6) -a(n-7) +a(n-8) +a(n-9) +a(n-10) -a(n-13) -a(n-14) +a(n-15). (End)

Mathematica

Table[Count[IntegerPartitions[n], {5, ___}], {n, 0, 55}] (* corrected by Harvey P. Dale, Oct 24 2011 *)
Table[Length[IntegerPartitions[n, {5}]], {n, 0, 55}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^5/Product[1 - x^k, {k, 1, 5}], {x, 0, 65}], x] (* Robert A. Russell, May 13 2018 *)
Drop[LinearRecurrence[{1, 1, 0, 0, -1, -1, -1, 1, 1, 1, 0, 0, -1, -1, 1}, Append[Table[0, {14}], 1], 110], 9] (* Robert A. Russell, May 17 2018 *)

Prog

(PARI)
a(n)=round((n^4+10*(n^3+n^2)-75*n-45*(-1)^n*n)/2880);
for(n=0, 10000, print(n, " ", a(n))); /* b-file format */
/* Washington Bomfim, Jul 03 2012 */
(PARI) x='x+O('x^99); concat(vector(5), Vec(x^5/prod(k=1, 5, 1-x^k))) \\ Altug Alkan, May 17 2018
(GAP) List([0..70], n->NrPartitions(n, 5)); # Muniru A Asiru, May 17 2018

Crossrefs

Cf. A026810, A026812, A026813, A026814, A026815, A026816, A002622 (partial sums), A008667 (first differences).

Sequence in context: A062684 A341912 A033485 * A001401 A347549 A008628

Adjacent sequences: A026808 A026809 A026810 * A026812 A026813 A026814

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from Robert G. Wilson v, Jan 11 2002

a(0)=0 inserted by Joerg Arndt, Jul 04 2012

Status

approved