A194543 -id:A194543 - cp's OEIS frontend

A025157 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 3.

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 58, 66, 73, 82, 91, 102, 113, 126, 139, 155, 171, 190, 209, 232, 255, 282, 310, 342, 375, 413, 452, 497, 544, 596, 651, 713, 778, 850, 927, 1011, 1101, 1200, 1305, 1420, 1544, 1677, 1821, 1977, 2144, 2324, 2519, 2728

Offset: 0

Clark Kimberling

nonn

Also number of partitions of n into distinct parts in which the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Aug 06 2004

			a(12) = 6 because we have 12 = 11+1 = 10+2 = 9+3 = 8+4 = 7+4+1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
I. Martinjak and D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.

Cf. A003114, A096401, A263719.

Column k=3 of A194543.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(n>
      ceil(i*(i+3)/6), 0, b(n, i-1)+b(n-i, min(n-i, i-3))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 26 2022

nn=50; CoefficientList[Series[Sum[x^(j(3j-1)/2)Product[1/(1-x^i), {i, 1, j}], {j, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 21 2013 *)

my(N=99, x='x+O('x^N)); Vec(sum(k=0, N, x^(k*(3*k-1)/2)/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 13 2022

G.f.: sum(i>=1, x^(3*A000217(i)-2*i)/product(j=1..i, 1-x^j)). - Jon Perry, Jul 20 2004
G.f.: sum(n>=0, x^(n*(3*n-1)/2)/prod(k=1..n,1-x^k)). - Joerg Arndt, Jan 29 2011
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*r*sqrt(Pi*(1+3*r^2))), where r = A263719 = ((9+sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9+sqrt(93))))^(1/3) = 0.682327803828019327369483739711048256891188581898... is the root of the equation r^3 + r = 1 and c = 3*(log(r))^2/2 + polylog(2, 1-r) = 0.566433354765746647188050807325058683443823543741343518... . - Vaclav Kotesovec, Jan 02 2016

Prepended a(0)=1, Joerg Arndt, Jul 21 2013

A025162 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 8.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 11, 12, 13, 15, 16, 18, 20, 22, 24, 27, 29, 32, 35, 38, 41, 45, 48, 52, 56, 60, 64, 69, 73, 78, 83, 89, 94, 101, 107, 115, 122, 131, 139, 150, 159, 171, 182, 196, 208, 224, 238, 256, 272, 292, 310

Offset: 1

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003114, A025157-A025162.

Column k=8 of A194543.

G.f.: Sum(x^(4*k^2-3*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

More terms from Naohiro Nomoto, Feb 27 2002

A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 25, 28, 31, 35, 38, 43, 47, 53, 58, 65, 71, 80, 87, 97, 106, 118, 128, 142, 154, 170, 185, 203, 220, 242, 262, 287, 311, 340, 368, 402, 435, 474, 513, 558, 603, 656, 708, 768, 829, 898, 968, 1048

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n such that if k is the largest part, then each 1,2,...,k-1 occur at least 4 times. Example: a(8)=3 because we have [2,2,1,1,1,1], [2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 17 2006

			a(8) = 3 because we have [8], [7,1] and [6,2].

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003114, A025157-A025162.

Column k=4 of A194543.

g:=sum(x^(2*k^2-k)/product(1-x^j,j=1..k),k=1..7): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..66); # Emeric Deutsch, Apr 17 2006

nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(j*(2*j - 1))/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 02 2016 *)

G.f.: Sum(x^(2*k^2-k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*sqrt(Pi*r^3*(1+4*r^3))), where r = 0.72449195900051561158837228218703656578649448135001101727... is the root of the equation r^4 + r = 1 and c = 2*log(r)^2 + polylog(2, 1-r) = 0.50498141294472195442598916817438524920370382784609501495065... . - Vaclav Kotesovec, Jan 02 2016

More terms from Vladeta Jovovic, Aug 12 2004

A025159 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 24, 26, 29, 32, 35, 38, 42, 46, 50, 55, 60, 66, 72, 79, 86, 95, 103, 113, 123, 135, 146, 160, 173, 189, 204, 222, 239, 260, 280, 303, 326, 353, 379, 410, 440, 475, 510, 550, 590, 636, 682

Offset: 1

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003114, A025157-A025162.

Column k=5 of A194543.

G.f.: Sum(x^(5/2*k^2-3/2*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

a(n) ~ c^(1/4) * r * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1-r)*(5-4*r)) * n^(3/4)), where r = 0.754877666246692760049508896358528691894606617772793143989... is the root of the equation r^5 + r = 1 and c = 5*log(r)^2/2 + polylog(2, 1-r) = 0.45973143655369174108251201834952526825516678... . - Vaclav Kotesovec, Jan 02 2016

More terms from Vladeta Jovovic, Aug 12 2004

A025160 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 25, 27, 30, 33, 36, 39, 43, 46, 50, 54, 59, 63, 69, 74, 81, 87, 95, 102, 112, 120, 131, 141, 154, 165, 180, 193, 210, 225, 244, 261, 283, 302, 326, 348, 375, 400, 430, 458, 492

Offset: 1

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003114, A025157-A025162.

Column k=6 of A194543.

G.f.: Sum(x^(3*k^2-2*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

More terms from Vladeta Jovovic, Aug 12 2004

A025161 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 12, 14, 15, 17, 19, 21, 23, 26, 28, 31, 34, 37, 40, 44, 47, 51, 55, 59, 63, 68, 73, 78, 84, 90, 97, 104, 112, 120, 130, 139, 150, 161, 174, 186, 201, 215, 232, 248, 267, 285, 307, 327, 351, 374, 401

Offset: 1

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003114, A025157-A025162.

Column k=7 of A194543.

G.f.: Sum(x^(7/2*k^2-5/2*k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

More terms from Naohiro Nomoto, Feb 27 2002

cp's OEIS Frontend

A025157 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 3.

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A025162 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 8.

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A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.

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A025159 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 5.

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A025160 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 6.

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Extensions

A025161 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 7.

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Links

Crossrefs

Formula

Extensions