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A035632
Number of partitions of n into parts 5k+1 and 5k+3 with at least one part of each type.
3
0, 0, 0, 1, 1, 1, 2, 2, 4, 5, 5, 7, 8, 11, 14, 15, 19, 22, 27, 33, 37, 44, 50, 59, 71, 79, 93, 106, 120, 142, 159, 181, 207, 232, 267, 301, 339, 383, 428, 486, 544, 609, 683, 758, 853, 951, 1056, 1180, 1304, 1453, 1616, 1785, 1980, 2185, 2417, 2674, 2947, 3253
OFFSET
1,7
LINKS
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 3))). - Robert Price, Aug 16 2020
MATHEMATICA
nmax = 58; s1 = Range[0, nmax/5]*5 + 1; s2 = Range[0, nmax/5]*5 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 58; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
KEYWORD
nonn
STATUS
approved