A035677 Number of partitions of n into parts 8k and 8k + 6 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 3, 0, 6, 0, 1, 0, 3, 0, 7, 0, 12, 0, 3, 0, 7, 0, 15, 0, 21, 0, 7, 0, 16, 0, 28, 0, 36, 0, 16, 0, 31, 0, 50, 0, 60, 0, 32, 0, 57, 0, 85, 0, 98, 0, 60, 0, 100, 0, 141, 0, 157, 0, 107, 0, 169, 0, 226, 0, 248, 0

Offset

1,22

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 361 terms from Antti Karttunen)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 6)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 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 {0, 3}, add(b(n-i*j, i-1,
      `if`(h=0, 1, t), `if`(h=3, 1, s)), j=1..n/i), 0))(irem(i, 4))))
    end:
a:= n-> `if`(n::odd, 0, b(n/2$2, 0$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 17 2020

Mathematica

nmax = 87; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 87; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 13 2020 *)

Prog

(PARI)
parts8katleast(up_to, n) = select(x -> (x>=n), vector(((up_to+0)>>3), k, ((k<<3)-0)));
parts8kplus6(up_to) = vector(((up_to+2)>>3), k, ((k<<3)-2));
partitions_for_A035677(n, parts, from=1, has8k6parts=0) = if(!n, (has8k6parts>0), my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_for_A035677(n-parts[i], parts, i, (has8k6parts+(6==(parts[i]%8)))))); (s));
A035677(n) = if(n%2, 0, sum(i=1, n>>3, my(k = i*8); partitions_for_A035677(n-k, vecsort(setunion(parts8katleast(n-k, k), parts8kplus6(n-k)), , 4)))); \\ Antti Karttunen, Feb 06 2019

Crossrefs

Cf. A035441-A035468, A035618-A035676, A035678-A035699.

Bisections give: A035623 (even part), A000004 (odd part).

Sequence in context: A083206 A069531 A335487 * A283470 A101941 A089313

Adjacent sequences: A035674 A035675 A035676 * A035678 A035679 A035680

Keyword

nonn

Author

Olivier Gérard

Status

approved