A066655 -id:A066655 - cp's OEIS frontend

A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.

1, 1, 2, 7, 23, 84, 331, 1367, 5812, 25331, 112804, 511045, 2348042, 10919414, 51313463, 243332340, 1163105227, 5598774334, 27119990519, 132107355553, 646793104859, 3181256110699, 15712610146876, 77903855239751, 387609232487489, 1934788962992123

Offset: 0

Reinhard Zumkeller, Feb 20 2010

nonn

a(n) is also the number of partitions of n^3 into n distinct parts <= n*(n+1). a(3) = 7: [4,11,12], [5,10,12], [6,9,12], [6,10,11], [7,8,12], [7,9,11], [8,9,10]. - Alois P. Heinz, Jan 25 2012

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..720 (terms 0..200 from Alois P. Heinz)

Cf. A066655, A097356, A258234.

Table[Length[IntegerPartitions[n(n + 1)/2, n]], {n, 10}] (* Alonso del Arte, Aug 12 2011 *)
Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)/2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

a(n)=
{
    local(tr=n*(n+1)/2, x='x+O('x^(tr+3)), gf);
    gf = 1 / prod(k=1,n, 1-x^k); /* g.f. for partitions into parts <=n */
    return( polcoeff( truncate(gf), tr ) );
} /* Joerg Arndt, Aug 14 2011 */

a(n) = A026820(A000217(n),n).

a(n) ~ c * d^n / n^2, where d = 5.4008719041181541524660911191042700520294... = A258234, c = 0.6326058791290010900659134913629203727... . - Vaclav Kotesovec, Sep 07 2014

More terms from D. S. McNeil, Aug 12 2011

A104383 Number of distinct partitions of triangular numbers n*(n+1)/2.

Original entry on oeis.org

1, 1, 2, 4, 10, 27, 76, 222, 668, 2048, 6378, 20132, 64234, 206848, 671418, 2194432, 7215644, 23853318, 79229676, 264288462, 884987529, 2973772212, 10024300890, 33888946600, 114872472064, 390334057172, 1329347719190, 4536808055808, 15513418629884

Offset: 0

Paul D. Hanna, Mar 04 2005

nonn

Equals row sums of triangle A104382. Asymptotics: a(n) ~ exp(Pi*sqrt((n^2+n)/6))/(2*6^(1/4))/(n^2+n)^(3/4).

Abramowitz, M. and Stegun, I. A. (Editors). "Partitions into Distinct Parts." S24.2.2 in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, pp. 825-826, 1972.

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Partition Function Q.

Cf. A000009, A000217, A066655, A104382.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n*(n+1)/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 24 2016

Join[{1},PartitionsQ/@Accumulate[Range[30]]] (* Harvey P. Dale, Dec 29 2012 *)

{a(n)=polcoeff(prod(k=1,n*(n+1)/2,1+x^k,1+x*O(x^(n*(n+1)/2))),n*(n+1)/2)}

Limit_{n-->inf} a(n+1)/a(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...

a(n) = A000009(A000217(n)). - Alois P. Heinz, Nov 24 2016

a(0)=1 prepended by Alois P. Heinz, Aug 05 2016

A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 33, 68, 144, 312, 686, 1523, 3405, 7652, 17284, 39246, 89552, 205253, 472297, 1090544, 2525904, 5867037, 13663248, 31896309, 74628130, 174972341, 411032475, 967307190, 2280248312, 5383723722, 12729879673, 30141755384, 71462883813

Offset: 0

Moshe Shmuel Newman, Feb 15 2007

nonn

Also the number of self-conjugate partitions of the n-th triangular number.

			The 5th triangular number is 15. Writing this as a sum of distinct odd numbers: 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3 are all the possibilities. So a(5) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Sequences A066655 and A104383 do the same thing for triangular numbers, with partitions or distinct partitions. Sequences A072213 and A072243 are analogs for squares rather than triangular numbers.

Cf. A000217.

g:= mul(1+x^(2*j+1),j=0..900): seq(coeff(g,x,n*(n+1)/2),n=0..40); # Emeric Deutsch, Feb 27 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i^2n, 0, b(n-2*i+1, i-1))))
    end:
a:= n-> b(n*(n+1)/2, ceil(n*(n+1)/4)*2-1):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 31 2018

a[n_] := SeriesCoefficient[QPochhammer[-x, x^2], {x, 0, n*(n+1)/2}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 25 2018 *)

More terms from Emeric Deutsch, Feb 27 2007

a(0)=1 prepended by Alois P. Heinz, Jan 31 2018

A267709 Number of partitions of pentagonal numbers.

Original entry on oeis.org

1, 1, 7, 77, 1002, 14883, 239943, 4087968, 72533807, 1327710076, 24908858009, 476715857290, 9275102575355, 182973889854026, 3652430836071053, 73653287861850339, 1498478743590581081, 30724985147095051099, 634350763653787028583, 13177726323474524612308

Offset: 0

Ilya Gutkovskiy, Apr 07 2016

nonn

			a(2) = 7, because second pentagonal number is a 5 and 5 can be partitioned in 7 distinct ways: 5, 4 + 1, 3 + 2, 3 + 1 + 1, 3 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..730 (terms 0..90 from Ilya Gutkovskiy)
Eric Weisstein's World of Mathematics, Partition, Partition Function P
Eric Weisstein's World of Mathematics, Pentagonal Number
Index entries for related partition-counting sequences

Cf. A000041, A000326, A066655, A072213.

Table[PartitionsP[n ((3 n - 1)/2)], {n, 0, 19}]

a(n)=numbpart(n*(3*n-1)/2) \\ Charles R Greathouse IV, Jul 26 2016

from sympy.ntheory import npartitions
print([npartitions(n*(3*n - 1)//2) for n in range(51)]) # Indranil Ghosh, Apr 11 2017

a(n) = A000041(A000326(n)).
a(n) ~ exp((Pi*sqrt(n*(3*n - 1)))/sqrt(3))/(2*sqrt(3)*n*(3*n - 1)).
a(n) = [x^(n*(3*n-1)/2)] Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017

A137736 Number of set partitions of [n*(n-1)/2].

Original entry on oeis.org

1, 1, 1, 5, 203, 115975, 1382958545, 474869816156751, 6160539404599934652455, 3819714729894818339975525681317, 139258505266263669602347053993654079693415, 359334085968622831041960188598043661065388726959079837

Offset: 0

Thomas Wieder, Feb 09 2008

nonn

Among n persons we have (n^2-n)/2 undirected relations. We can set partition these relations into (up to) a(n) = Bell((n^2-n)/2) sets.
The number of graphs on n labeled nodes is A006125(n) = Sum_{k=0..(n^2-n)/2} binomial((n^2-n)/2,k).
See also A066655 which equals A066655(n) = Sum_{k=0..(n^2-n)/2} P((n^2-n)/2,k) where P(n) is the number of integer partitions of n.
See also A135084 = A000110(2^n-1) and A135085 = A000110(2^n).

			a(4) = Bell(6) = 203.

Cf. A000110, A006125, A066655, A135084, A135085, A161680.

seq(combinat[bell](n*(n-1)/2), n=0..12);

a[n_]=BellB[n(n-1)/2];Array[a,12,0] (* James C. McMahon, Jun 02 2025 *)

a(n) = Bell(n*(n-1)/2) = A000110(n*(n-1)/2).

a(n) = Sum_{k=0..(n^2-n)/2} Stirling2((n^2-n)/2,k).

a(0)=1 prepended by Alois P. Heinz, Jul 24 2024

A336605 a(n) is the number of partitions of the n-th tetrahedral number (A000292).

Original entry on oeis.org

1, 1, 5, 42, 627, 14883, 526823, 26543660, 1844349560, 172389800255, 21248279009367, 3397584011986773, 695143713458946040, 179855916453958267598, 58248417552751868050007, 23402165235974892374954302, 11571309261543787320061392679

Offset: 0

Robert Bilinski, Sep 13 2020

nonn

Cf. A000041, A000292, A066655.

a(n) = numbpart(n*(n+1)*(n+2)/6); \\ Michel Marcus, Sep 14 2020

a(n) = p(n*(n+1)*(n+2)/6).

a(n) = A000041(A000292(n)).

cp's OEIS Frontend

A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.

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A104383 Number of distinct partitions of triangular numbers n*(n+1)/2.

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A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

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A267709 Number of partitions of pentagonal numbers.

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A137736 Number of set partitions of [n*(n-1)/2].

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A336605 a(n) is the number of partitions of the n-th tetrahedral number (A000292).

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