A173519 -id:A173519 - cp's OEIS frontend

A026820 Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <= k for 1 <= k <= n. Also number of partitions of n into at most k parts.

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 11, 13, 14, 15, 1, 5, 10, 15, 18, 20, 21, 22, 1, 5, 12, 18, 23, 26, 28, 29, 30, 1, 6, 14, 23, 30, 35, 38, 40, 41, 42, 1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56, 1, 7, 19, 34, 47, 58, 65, 70, 73, 75, 76, 77

Offset: 1

Clark Kimberling

			Triangle starts:
  1;
  1, 2;
  1, 2,  3;
  1, 3,  4,  5;
  1, 3,  5,  6,  7;
  1, 4,  7,  9, 10, 11;
  1, 4,  8, 11, 13, 14, 15;
  1, 5, 10, 15, 18, 20, 21, 22;
  1, 5, 12, 18, 23, 26, 28, 29, 30;
  1, 6, 14, 23, 30, 35, 38, 40, 41, 42;
  1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56;
  ...

G. Chrystal, Algebra, Vol. II, p. 558.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

Alois P. Heinz, Robert G. Wilson v, Rows n = 1..141, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831. [scanned copy]
Carolyn Echter, Georg Maier, Juan-Diego Urbina, Caio Lewenkopf, and Klaus Richter, Many-body density of states of bosonic and fermionic gases: a combinatorial approach, arXiv:2409.08696 [cond-mat.quant-gas], 2024. See p. 10.
L. Euler, Introductio in Analysin Infinitorum, Book I, chapter XVI.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers.
R. Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, Univ. Wien, 2013.
Sergei Viznyuk, C program.
Sergei Viznyuk, Local copy of C program.
Eric Weisstein's World of Mathematics, Partition Function q.
Index entries for sequences related to partitions - _Reinhard Zumkeller_, Jan 21 2010

Partial sums of rows of A008284, row sums give A058397, central terms give A171985, mirror is A058400.
T(n,n) = A000041(n), T(n,1) = A000012(n), T(n,2) = A008619(n) for n>1, T(n,3) = A001399(n) for n>2, T(n,4) = A001400(n) for n>3, T(n,5) = A001401(n) for n>4, T(n,6) = A001402(n) for n>5, T(n,7) = A008636(n) for n>6, T(n,8) = A008637(n) for n>7, T(n,9) = A008638(n) for n>8, T(n,10) = A008639(n) for n>9, T(n,11) = A008640(n) for n>10, T(n,12) = A008641(n) for n>11, T(n,n-2) = A007042(n-1) for n>2, T(n,n-1) = A000065(n) for n>1.
Cf. A008284, A026840, A134737, A173519.

import Data.List (inits)
a026820 n k = a026820_tabl !! (n-1) !! (k-1)
a026820_row n = a026820_tabl !! (n-1)
a026820_tabl = zipWith
   (\x -> map (p x) . tail . inits) [1..] $ tail $ inits [1..] where
   p 0 _ = 1
   p _ [] = 0
   p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 18 2013

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Apr 21 2012

t[n_, k_] := Length@ IntegerPartitions[n, k]; Table[ t[n, k], {n, 12}, {k, n}] // Flatten
(* Second program: *)
T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k]]]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)

T(n,k)=my(s); forpart(v=n,s++,,k); s \\ Charles R Greathouse IV, Feb 27 2018

from sage.combinat.partition import number_of_partitions_length
from itertools import accumulate
for n in (1..11):
    print(list(accumulate([number_of_partitions_length(n, k) for k in (1..n)])))
# Peter Luschny, Jul 28 2022

T(T(n,n),n) = A134737(n). - Reinhard Zumkeller, Nov 07 2007
T(A000217(n),n) = A173519(n). - Reinhard Zumkeller, Feb 20 2010
T(n,k) = T(n,k-1) + T(n-k,k). - Thomas Dybdahl Ahle, Jun 13 2011
T(n,k) = Sum_{i=1..min(k,floor(n/2))} T(n-i,i) + Sum_{j=1+floor(n/2)..k} A000041(n-j). - Bob Selcoe, Aug 22 2014 [corrected by Álvar Ibeas, Mar 15 2018]
O.g.f.: Product_{i>=0} 1/(1-y*x^i). - Geoffrey Critzer, Mar 11 2012
T(n,k) = A008284(n+k,k). - Álvar Ibeas, Jan 06 2015

A206226 Number of partitions of n^2 into parts not greater than n.

Original entry on oeis.org

1, 1, 3, 12, 64, 377, 2432, 16475, 116263, 845105, 6292069, 47759392, 368379006, 2879998966, 22777018771, 181938716422, 1465972415692, 11902724768574, 97299665768397, 800212617435074, 6617003142869419, 54985826573015541, 458962108485797208, 3846526994743330075

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Also the number of partitions of n^2 using n or fewer numbers. Thus for n=3 one has: 9; 1,8; 2,7; 3,6; 4,5; 1,1,7; 1,2,6; 1,3,5; 1,4,4; 2,2,5; 2,3,4; 3,3,3. - J. M. Bergot, Mar 26 2014 [computations done by Charles R Greathouse IV]
The partitions in the comments above are the conjugates of the partitions in the definition. By conjugation we have: "partitions into parts <= m" are equinumerous with "partitions into at most m parts". - Joerg Arndt, Mar 31 2014
From Vaclav Kotesovec, May 25 2015: (Start)
In general, "number of partitions of j*n^2 into parts that are at most n" is (for j>0) asymptotic to c(j) * d(j)^n / n^2, where c(j) and d(j) are a constants.
-------
j  c(j)
1  0.1582087202672504149766310999238...
2  0.0794245035465730707705885572860...
3  0.0530017980244665552354063060738...
4  0.0397666338404544208556554596295...
5  0.0318193213988281353709268311928...
...
17 0.0093617308583114626385718275875...
   c(j) for big j asymptotically approaches 1 / (2*Pi*j).
---------
j    d(j)
1    9.15337019245412246194853029240... = A258268
2   16.57962120993269533568313969522...
3   23.98280768122086592445663786762...
4   31.37931997386325137074644287711...
5   38.77298550971449870728474612568...
...
17 127.45526806942537991146993713837...
   d(j) for big j asymptotically approaches j * exp(2).
(End)
d(j) = r^(2*j+1)/(r-1), where r is the root of the equation polylog(2, 1-r) + (j+1/2)*log(r)^2 = 0. - Vaclav Kotesovec, Jun 11 2015

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206227, A206240, A107379, A258268.
Column k=2 of A238016.
Cf. A258296 (j=2), A258293 (j=3), A258294 (j=4), A258295 (j=5).

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n^2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)
(* A program to compute the constants d(j) *) Table[r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->60], {j, 1, 5}] (* Vaclav Kotesovec, Jun 11 2015 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2)))),n^2)}
for(n=0,25,print1(a(n),", "))

a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.1582087202672504149766310999238742... . - Vaclav Kotesovec, Sep 07 2014

A206240 Number of partitions of n^2-n into parts not greater than n.

Original entry on oeis.org

1, 1, 2, 7, 34, 192, 1206, 8033, 55974, 403016, 2977866, 22464381, 172388026, 1341929845, 10573800028, 84192383755, 676491536028, 5479185281572, 44692412971566, 366844007355202, 3028143252035976, 25123376972033392, 209401287806758273, 1752674793617241002

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Also the number of partitions of n^2 into exactly n parts. - Seiichi Manyama, May 07 2018

			From _Seiichi Manyama_, May 07 2018: (Start)
n | Partitions of n^2 into exactly n parts
--+-------------------------------------------------------
1 | 1.
2 | 3+1 = 2+2.
3 | 7+1+1 = 6+2+1 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 3+3+3. (End)

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206226, A206227, A107379, A258268.

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2-n, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n-1)}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2-n)))),n^2-n)}
for(n=0,30,print1(a(n),", "))

a(n) = [x^(n^2-n)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.153370192454122461948530292401354540073... = A258268, c = 0.07005383646855329845970382163053268... . - Vaclav Kotesovec, Sep 07 2014

A258268 Decimal expansion of a constant related to A206226.

Original entry on oeis.org

9, 1, 5, 3, 3, 7, 0, 1, 9, 2, 4, 5, 4, 1, 2, 2, 4, 6, 1, 9, 4, 8, 5, 3, 0, 2, 9, 2, 4, 0, 1, 3, 5, 4, 5, 4, 0, 0, 7, 3, 3, 2, 7, 2, 0, 4, 1, 2, 1, 8, 4, 8, 8, 4, 9, 6, 8, 9, 2, 6, 3, 2, 0, 1, 4, 7, 6, 1, 3, 8, 3, 7, 6, 6, 8, 9, 5, 7, 3, 1, 6, 2, 3, 9, 1, 5, 1, 9, 0, 2, 5, 5, 8, 7, 9, 5, 1, 9, 2, 8, 4, 5, 3, 8, 9

Offset: 1

Vaclav Kotesovec, May 25 2015

			9.153370192454122461948530292401354540073...

Cf. A206226, A206227, A206240, A258234, A107379, A173519, A238016, A258789.

r^3/(r-1) /.FindRoot[-PolyLog[2, 1-r] == 3*Log[r]^2/2, {r, E}, WorkingPrecision->120] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A206226(n)^(1/n).
Equals limit n->infinity A206227(n)^(1/n).
Equals limit n->infinity A206240(n)^(1/n).

More digits from Vaclav Kotesovec, Jun 10 2015

A321470 Number of integer partitions of the n-th triangular number 1 + 2 + ... + n that can be obtained by choosing a partition of each integer from 1 to n and combining.

Original entry on oeis.org

1, 1, 2, 5, 16, 54, 212, 834, 3558, 15394, 69512, 313107, 1474095, 6877031, 32877196

Offset: 0

Gus Wiseman, Nov 11 2018

a(n) is the number of integer partitions finer than (n, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + n ordered by refinement.

a(n+1)/a(n) appears to converge as n -> oo. - Chai Wah Wu, Nov 14 2018

			The a(1) = 1 through a(4) = 16 partitions:
  (1)  (21)   (321)     (4321)
       (111)  (2211)    (32221)
              (3111)    (33211)
              (21111)   (42211)
              (111111)  (43111)
                        (222211)
                        (322111)
                        (331111)
                        (421111)
                        (2221111)
                        (3211111)
                        (4111111)
                        (22111111)
                        (31111111)
                        (211111111)
                        (1111111111)
The partition (222211) is the combination of (22)(21)(2)(1), so is counted under a(4). The partition (322111) is the combination of (22)(3)(11)(1), (31)(21)(2)(1), or (211)(3)(2)(1), so is also counted under a(4).

Cf. A000217, A001970, A002846, A063834, A066723, A173519, A213427, A242422, A261049, A265947, A271619, A299201, A300383, A317141.

Cf. A321467, A321468, A321471, A321472, A321514.

Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@Range[1,n]]]],{n,6}]

from collections import Counter
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A321470_gen(): # generator of terms
    aset = {(1,)}
    yield 1
    for n in count(2):
        yield len(aset)
        aset = {tuple(sorted(p+q)) for p in aset for q in (tuple(sorted(Counter(q).elements())) for q in partitions(n))}
A321470_list = list(islice(A321470_gen(),10)) # Chai Wah Wu, Sep 20 2023

a(n) <= A173519(n). - David A. Corneth, Sep 20 2023

a(9)-a(11) from Alois P. Heinz, Nov 12 2018
a(12)-a(13) from Chai Wah Wu, Nov 13 2018
a(14) from Chai Wah Wu, Sep 20 2023

A066655 Number of partitions of n*(n-1)/2.

Original entry on oeis.org

1, 1, 3, 11, 42, 176, 792, 3718, 17977, 89134, 451276, 2323520, 12132164, 64112359, 342325709, 1844349560, 10015581680, 54770336324, 301384802048, 1667727404093, 9275102575355, 51820051838712, 290726957916112, 1637293969337171, 9253082936723602

Offset: 1

Roberto E. Martinez II, Jan 10 2002

nonn

Number of partitions of the number of edges of the complete graph of order n, K_n.

			a(4) = p(6) = 11.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000041, A000217, A007294, A104383.

Cf. A173519. - Reinhard Zumkeller, Feb 20 2010

Table[PartitionsP[n(n-1)/2], {n, 1, 30}]

combinat::partitions::count(binomial(n+2,n)) $n=-1..40 // Zerinvary Lajos, Apr 16 2007

a(n) = numbpart(n*(n-1)/2); \\ Michel Marcus, Dec 18 2017

a(n) = p(n*(n-1)/2) = A000041(n*(n-1)/2).
a(n) ~ exp(Pi*sqrt(n*(n-1)/3))/(2*sqrt(3)*n*(n - 1)). - Ilya Gutkovskiy, Jan 13 2017
a(n) ~ exp(Pi*(n - 1/2) / sqrt(3)) / (2*sqrt(3)*n^2). - Vaclav Kotesovec, May 17 2018

More terms from Vladeta Jovovic, Jan 12 2002

Edited by Dean Hickerson, Jan 14 2002

A258234 Decimal expansion of a constant related to A107379.

Original entry on oeis.org

5, 4, 0, 0, 8, 7, 1, 9, 0, 4, 1, 1, 8, 1, 5, 4, 1, 5, 2, 4, 6, 6, 0, 9, 1, 1, 1, 9, 1, 0, 4, 2, 7, 0, 0, 5, 2, 0, 2, 9, 4, 3, 7, 7, 1, 0, 1, 9, 1, 6, 7, 0, 5, 7, 0, 9, 3, 1, 7, 0, 6, 0, 1, 4, 4, 8, 4, 4, 8, 5, 1, 5, 9, 5, 0, 7, 5, 8, 1, 7, 7, 8, 9, 8, 8, 7, 4, 7, 9, 2, 0, 0, 0, 0, 6, 2, 0, 6, 2, 7, 7, 6, 7, 0, 0

Offset: 1

Vaclav Kotesovec, May 24 2015

Limit n->infinity (Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx)^(1/n) = Limit n->infinity (A258191(n)/A258192(n))^(1/n) = 1/A258234 = 0.18515528932235959464731321119795428527382236445907508398560553036... .

			5.4008719041181541524660911191042700520294...

StackExchange - Mathematica, No response to an infinite limit

Cf. A107379, A173519, A258191, A258192, A258268, A258788.

r^2/(r-1) /.FindRoot[-PolyLog[2, 1-r] == Log[r]^2, {r, E}, WorkingPrecision->117] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A107379(n)^(1/n).

Equals limit n->infinity A173519(n)^(1/n).

More terms from Vaclav Kotesovec, Jun 09 2015

A320932 a(n) = [x^(n(n+1)/2)] Product_{k=1..n} Sum_{m>=0} x^(km^2).

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 20, 51, 141, 381, 1001, 2796, 7861, 22306, 64129, 185692, 540468, 1585246, 4674464, 13846636, 41216933, 123176849, 369410571, 1111661833, 3355466306, 10156304314, 30821794651, 93761053797, 285859742756, 873355481467, 2673455511946, 8198687383812

Offset: 0

Seiichi Manyama, Oct 28 2018

nonn

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n*(n+1)/2.

			1*1^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*1^2 = 15.
1*2^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*1^2 = 15.
1*0^2 + 2*2^2 + 3*1^2 + 4*1^2 + 5*0^2 = 15.
1*3^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*0^2 = 15.
1*1^2 + 2*1^2 + 3*2^2 + 4*0^2 + 5*0^2 = 15.
1*2^2 + 2*2^2 + 3*1^2 + 4*0^2 + 5*0^2 = 15.
So a(5) = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..300 (first 101 terms from Seiichi Manyama)

Cf. A000122, A000217, A010052, A173519, A300446, A320931.

b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
    end:
a:= n-> b(n*(n+1)/2, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 28 2018

nmax = 30; Table[SeriesCoefficient[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, n}], {x, 0, n*(n+1)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

{a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n*(n+1)\(2*i)), x^(i*j^2)+x*O(x^(n*(n+1)/2)))), n*(n+1)/2)}

a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.

A206227 Number of partitions of n^2+n into parts not greater than n.

Original entry on oeis.org

1, 1, 4, 19, 108, 674, 4494, 31275, 225132, 1662894, 12541802, 96225037, 748935563, 5900502806, 46976736513, 377425326138, 3056671009814, 24930725879856, 204623068332997, 1688980598900228, 14012122025369431, 116784468316023069, 977437078888272796, 8212186058546599006

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206226, A206240, A107379, A258268.

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2+n, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2+n)))),n^2+n)}
for(n=0,30,print1(a(n),", "))

a(n) = [x^(n^2+n)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.3572966225745094270279188015952797... . - Vaclav Kotesovec, Sep 07 2014

A320931 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 12, 24, 80, 292, 966, 3876, 15554, 61608, 254612, 1065676, 4471672, 19074968, 82043172, 354365492, 1543432514, 6760146292, 29732837780, 131440491584, 583419967664, 2598585783488, 11615321544700, 52079369904384, 234157152231726, 1055628140278948, 4770576024205060

Offset: 0

Seiichi Manyama, Oct 28 2018

nonn

Also the number of integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n*(n+1)/2.

			Solutions (a_1, a_2, ... , a_4) to the equation a_1^2 + 2*a_2^2 + ... + 4*a_4^2 = 10.
-------------------------------------------------------------------------------------
( 1,  1,  1,  1), ( 1,  1,  1, -1),
( 1,  1, -1,  1), ( 1,  1, -1, -1),
( 1, -1,  1,  1), ( 1, -1,  1, -1),
( 1, -1, -1,  1), ( 1, -1, -1, -1),
(-1,  1,  1,  1), (-1,  1,  1, -1),
(-1,  1, -1,  1), (-1,  1, -1, -1),
(-1, -1,  1,  1), (-1, -1,  1, -1),
(-1, -1, -1,  1), (-1, -1, -1, -1),
( 2,  1,  0,  1), ( 2,  1,  0, -1),
( 2, -1,  0,  1), ( 2, -1,  0, -1),
(-2,  1,  0,  1), (-2,  1,  0, -1),
(-2, -1,  0,  1), (-2, -1,  0, -1).

Vaclav Kotesovec, Table of n, a(n) for n = 0..300 (first 101 terms from Seiichi Manyama)

Cf. A000122, A000217, A173519, A320067, A320932.

nmax = 25; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n*(n+1)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

a(n) ~ c * d^n / n^(7/4), where d = 4.818071572655... and c = 0.5869031198... - Vaclav Kotesovec, Oct 29 2018

cp's OEIS Frontend

A026820 Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <= k for 1 <= k <= n. Also number of partitions of n into at most k parts.

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A206226 Number of partitions of n^2 into parts not greater than n.

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A206240 Number of partitions of n^2-n into parts not greater than n.

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A258268 Decimal expansion of a constant related to A206226.

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A321470 Number of integer partitions of the n-th triangular number 1 + 2 + ... + n that can be obtained by choosing a partition of each integer from 1 to n and combining.

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A066655 Number of partitions of n*(n-1)/2.

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A258234 Decimal expansion of a constant related to A107379.

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A320932 a(n) = [x^(n(n+1)/2)] Product_{k=1..n} Sum_{m>=0} x^(km^2).