A053802 Number of basis partitions of n+49 with Durfee square size 7.

1, 2, 4, 8, 14, 24, 40, 64, 98, 148, 218, 314, 446, 622, 854, 1158, 1550, 2050, 2684, 3478, 4464, 5682, 7172, 8982, 11170, 13796, 16928, 20648, 25040, 30198, 36234, 43262, 51410, 60824, 71656, 84074, 98266, 114430, 132780, 153556, 177008

Offset: 0

Author

James Sellers, Mar 27 2000

Keywords

• easy
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• M. D. Hirschhorn, Basis partitions and Rogers-Ramanujan partitions, Discrete Math. 205 (1999), 241-243.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-1,-1,0,1,1,2,0,0,0,-2,-1,-1,0,1,1,0,1,0,0,-1,-1,1).

Programs

Mathematica

CoefficientList[Series[Times@@(1+x^Range[7])/Times@@(1-x^Range[7]),{x,0,40}],x] (* Harvey P. Dale, Aug 30 2017 *)

PARI

Vec((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)*(1+x^6)*(1+x^7) / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)) + O(x^40)) \\ Colin Barker, Jan 02 2020

Formula

G.f.: (1+q)(1+q^2)(1+q^3)(1+q^4)(1+q^5)(1+q^6)(1+q^7)/((1-q)(1-q^2)(1-q^3)(1-q^4)(1-q^5)(1-q^6)(1-q^7)).

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-8) + a(n-10) + a(n-11) + 2*a(n-12) - 2*a(n-16) - a(n-17) - a(n-18) + a(n-20) + a(n-21) + a(n-23) - a(n-26) - a(n-27) + a(n-28) for n>28. - Colin Barker, Jan 02 2020