A035676 -id:A035676 - cp's OEIS frontend

A035675 Number of partitions of n into parts 8k and 8k+4 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 11, 0, 0, 0, 22, 0, 0, 0, 25, 0, 0, 0, 44, 0, 0, 0, 51, 0, 0, 0, 83, 0, 0, 0, 98, 0, 0, 0, 149, 0, 0, 0, 177, 0, 0, 0, 259, 0, 0, 0, 309, 0, 0, 0, 436, 0, 0, 0, 521, 0, 0, 0, 716, 0, 0, 0, 857, 0, 0

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035674, A035676-A035699.

nmax = 90; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 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 = 90; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 13 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020

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

Original entry on oeis.org

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

Olivier Gérard

nonn

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

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

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

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

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 *)

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

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

A035445 Number of partitions of n into parts 8k or 8k+5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 1, 2, 0, 2, 0, 1, 4, 0, 2, 3, 1, 5, 0, 2, 7, 1, 5, 5, 2, 9, 1, 5, 12, 2, 10, 8, 5, 17, 2, 10, 20, 5, 19, 13, 10, 29, 5, 20, 32, 10, 34, 20, 20, 49, 10, 36, 50, 20, 59, 32, 37, 78, 20, 64, 77, 37, 97, 50, 66, 124, 37, 107, 117, 67, 157, 79

Offset: 1

Olivier Gérard

nonn

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

Cf. A035676.

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+8))*(1 - x^(8k+5))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 50; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8}~Join~{Range[0, kmax]*8 + 5}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(5/8) / (2^(29/16) * 3^(9/16) * Pi^(3/8) * n^(17/16)). - Vaclav Kotesovec, Aug 26 2015

cp's OEIS Frontend

A035675 Number of partitions of n into parts 8k and 8k+4 with at least one part of each type.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A035445 Number of partitions of n into parts 8k or 8k+5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula