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A035633
Number of partitions of n into parts 5k+1 and 5k+4 with at least one part of each type.
3
0, 0, 0, 0, 1, 1, 1, 1, 2, 4, 4, 4, 5, 7, 10, 11, 12, 14, 18, 23, 26, 29, 33, 40, 48, 55, 61, 70, 82, 96, 108, 121, 137, 158, 179, 202, 226, 255, 288, 325, 363, 406, 453, 508, 566, 632, 701, 781, 867, 963, 1066, 1182, 1306, 1445, 1592, 1759, 1939, 2139, 2350
OFFSET
1,9
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert Price)
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 4))). - Robert Price, Aug 16 2020
MATHEMATICA
nmax = 59; s1 = Range[0, nmax/5]*5 + 1; s2 = Range[0, nmax/5]*5 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 59; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
KEYWORD
nonn
STATUS
approved