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
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).
Crossrefs
Programs
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GAP
List([0..70],n->NrPartitions(n,5)); # Muniru A Asiru, May 17 2018
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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 *) -
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 */
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PARI
x='x+O('x^99); concat(vector(5), Vec(x^5/prod(k=1, 5, 1-x^k))) \\ Altug Alkan, May 17 2018
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)
Extensions
More terms from Robert G. Wilson v, Jan 11 2002
a(0)=0 inserted by Joerg Arndt, Jul 04 2012
Comments