A035648 -id:A035648

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A035647

Number of partitions of n into parts 6k+2 and 6k+4 with at least one part of each type.

+10
3

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 5, 0, 7, 0, 11, 0, 14, 0, 19, 0, 26, 0, 33, 0, 43, 0, 55, 0, 70, 0, 88, 0, 111, 0, 137, 0, 170, 0, 208, 0, 256, 0, 311, 0, 378, 0, 456, 0, 551, 0, 658, 0, 790, 0, 940, 0, 1119, 0, 1325, 0, 1570, 0, 1847, 0, 2179, 0, 2554, 0, 2996, 0, 3499

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,10

LINKS

Robert Price, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 4))). - Robert Price, Aug 16 2020

MATHEMATICA

nmax = 74; s1 = Range[0, nmax/6]*6 + 2; s2 = Range[0, nmax/6]*6 + 4;

Table[Count[IntegerPartitions[n, All, s1~Join~s2],

x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)

nmax = 74; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)

CROSSREFS

Cf. A035441-A035468, A035618-A035646, A035648-A035699.

Bisections give A035620 (even part), A000004 (odd part).

KEYWORD

nonn

AUTHOR

Olivier Gérard

STATUS

approved

A035649

Number of partitions of n into parts 6k+3 and 6k+4 with at least one part of each type.

+10
3

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 3, 1, 1, 3, 3, 1, 7, 3, 3, 8, 8, 3, 14, 9, 8, 16, 17, 9, 27, 19, 18, 32, 34, 20, 49, 40, 37, 58, 63, 43, 87, 74, 70, 104, 113, 82, 149, 135, 128, 177, 195, 152, 249, 232, 224, 298, 327, 266, 407, 392, 380, 485, 535, 455, 654, 639, 628

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,13

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 125 terms from Robert Price)

FORMULA

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 4))). - Robert Price, Aug 16 2020

MAPLE

b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,

b(n, i-1, t, s)+(h-> `if`(h in {3, 4}, add(b(n-i*j, i-1,

`if`(h=3, 1, t), `if`(h=4, 1, s)), j=1..n/i), 0))(irem(i, 6))))

end:

a:= n-> b(n$2, 0$2):

seq(a(n), n=1..75); # Alois P. Heinz, Aug 14 2020

MATHEMATICA

nmax = 69; s1 = Range[0, nmax/6]*6 + 3; s2 = Range[0, nmax/6]*6 + 4;

Table[Count[IntegerPartitions[n, All, s1~Join~s2],

x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 14 2020 *)

nmax = 69; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)