A035623 -id:A035623 - cp's OEIS frontend

A035622 Number of partitions of n into parts 4k and 4k+2 with at least one part of each type.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 4, 0, 10, 0, 11, 0, 22, 0, 25, 0, 44, 0, 51, 0, 83, 0, 98, 0, 149, 0, 177, 0, 259, 0, 309, 0, 436, 0, 521, 0, 716, 0, 857, 0, 1151, 0, 1376, 0, 1816, 0, 2170, 0, 2818, 0, 3361, 0, 4309, 0, 5132, 0, 6502, 0, 7728, 0, 9695, 0, 11501, 0, 14298

Offset: 0

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 101 terms from Robert Price)

Bisections give: A006477 (even part), A000004 (odd part).

Cf. A035441-A035468, A035618-A035621, A035623-A035699.

nmax = 70; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 0, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

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

A035624 Number of partitions of n into parts 4k+1 and 4k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 8, 8, 14, 15, 22, 23, 34, 37, 51, 54, 74, 81, 107, 116, 150, 165, 210, 229, 287, 316, 392, 430, 526, 580, 704, 774, 929, 1024, 1223, 1347, 1593, 1756, 2068, 2278, 2663, 2933, 3416, 3762, 4355, 4793, 5529, 6084, 6985, 7680, 8789

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert Price)

Cf. A035441-A035468, A035618-A035623, A035625-A035699.

nmax = 53; s1 = Range[0, nmax/4]*4 + 1; s2 = Range[0, nmax/4]*4 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 53; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

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

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A035622 Number of partitions of n into parts 4k and 4k+2 with at least one part of each type.

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A035624 Number of partitions of n into parts 4k+1 and 4k+2 with at least one part of each type.

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

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