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A035678 Number of partitions of n into parts 8k and 8k+7 with at least one part of each type. 2
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 3, 6, 0, 0, 0, 0, 1, 3, 7, 11, 0, 0, 0, 1, 3, 7, 14, 18, 0, 0, 1, 3, 7, 15, 25, 29, 0, 1, 3, 7, 15, 28, 43, 44, 1, 3, 7, 15, 29, 50, 69, 67, 3, 7, 15, 29, 53, 84, 110, 99, 7, 15, 29, 54, 91, 138, 168 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,23

LINKS

Robert Israel, Table of n, a(n) for n = 1..5000

FORMULA

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 7)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020

MAPLE

np:= combinat:-numbpart:

NP:= proc(n, m) if m > n then np(n) else np(n, m) fi end proc;

f:= proc(n) local r0;

   r0:= (-n) mod 8;

   add(np(s)*add(NP((n-8*s-7*r)/8, r), r=r0 .. floor((n-8*s)/7), 8), s=1..floor((n-1)/8))

end proc:

seq(f(n), n=1..100); # Robert Israel, Apr 06 2016

MATHEMATICA

nmax = 86; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 7;

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

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

nmax = 86; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 13 2020 *)

CROSSREFS

Cf. A035441-A035468, A035618-A035677, A035679-A035699.

Sequence in context: A269250 A334740 A059484 * A094518 A123074 A095269

Adjacent sequences:  A035675 A035676 A035677 * A035679 A035680 A035681

KEYWORD

nonn

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified May 19 18:35 EDT 2022. Contains 353847 sequences. (Running on oeis4.)