A091360 -id:A091360 - cp's OEIS frontend

A096597 Triangle read by rows: T[n,m] = number of plane partitions of n whose 3-dimensional Ferrers plot just fits inside an m X m X m box, i.e., with Max[parts, rows, columns] = m.

1, 0, 3, 0, 3, 3, 0, 4, 6, 3, 0, 3, 12, 6, 3, 0, 3, 21, 15, 6, 3, 0, 1, 31, 30, 15, 6, 3, 0, 1, 42, 60, 33, 15, 6, 3, 0, 0, 54, 102, 69, 33, 15, 6, 3, 0, 0, 64, 175, 132, 72, 33, 15, 6, 3, 0, 0, 73, 270, 246, 141, 72, 33, 15, 6, 3, 0, 0, 81, 417, 432, 276, 144, 72, 33, 15, 6, 3, 0, 0, 83

Offset: 1

Wouter Meeussen, Aug 14 2004

Row sums equal A000219 (plane partitions).
Conjecture: the last (floor(n/2)) terms of each row read backwards are 3*A091360 (partial sums of A000219).
Björner & Stanley (2010) give in eq.(3.7) MacMahon's generating function pp(r,s,t) for the number of plane partitions with rows <= r, columns <= s, parts <= t. For r = s = t = m, it simplifies to the g.f. f(m) given in formula. A g.f. for column m of this table is then f(m) - f(m-1). - M. F. Hasler, Sep 26 2018

			The table starts:
  n : T[n,1..n]
  1 : [1]
  2 : [0, 3]
  3 : [0, 3,  3]
  4 : [0, 4,  6,   3]
  5 : [0, 3, 12,   6,  3]
  6 : [0, 3, 21,  15,  6,  3]
  7 : [0, 1, 31,  30, 15,  6,  3]
  8 : [0, 1, 42,  60, 33, 15,  6, 3]
  9 : [0, 0, 54, 102, 69, 33, 15, 6, 3]
etc.
T[5,2] = 3 counts the plane partitions {{2,1},{2}}, {{2,1},{1,1}} and {{2,2},{1}}.

George E. Andrews, On a Partition Function of Richard Stanley, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), #R1.
A. Björner and R. P. Stanley, with A combinatorial miscellany, L'Enseignement Math., Monograph No. 42, 2010.
M. F. Hasler, A096597, rows 1..50, Sep 26 2018.

Cf. A096272, A091360.

(* see A089924 for "planepartitions[]" *) Table[Rest@CoefficientList[Plus@@(x ^ Max[Flatten[ # ], Length[ # ], Max[Length/@# ]]&/@ planepartitions[n]), x], {n, 19}]

A096597_row(n,c=vector(n))={for(i=1,#n=PlanePartitions(n),c[vecmax([#n[i], #n[i][1], n[i][1][1]])]++);c} \\ See A091298 for PlanePartitions().
{A096597(n,m,x=(O('x^n)+1)*'x,f(r)=prod(k=1,2*r-1,((1-x^(k+r))/(1-x^k))^min(k,2*r-k)))=polcoeff(f(m)-f(m-1),n)} \\ Replace "polcoeff(...,n)" by "Vec(...)" to get the whole column m up to row n (for "Vec(...,-n)", padded with leading 0's). - M. F. Hasler, Sep 26 2018

k-th column is CoefficientList[Series[qMacMahon[k]-qMacMahon[k-1], {q, 0, 3^k}], q] with qMacMahon[n_Integer]:=Product[qan[i+j+k-1]/qan[i+j+k-2], {i, n}, {j, n}, {k, n}] and qan[n_]:=(q^n-1)/(q-1). - Wouter Meeussen, Aug 28 2004
From M. F. Hasler, Sep 26 2018: (Start)
G.f. of column m: f(m)-f(m-1), where f(m) = Product_{k=1..2*m-1} ((1-X^(k+m))/(1-X^k))^min(k,2*m-k).
From the definition, we have T[n,m] = 0 if n > m^3.
Columns and reversed rows converge to 3*A091360: T[m+k,m] = T[2m,2m-k] = 3*A091360(k) for 0 <= k < m-1. (End)

Edited by M. F. Hasler, Sep 24 2018

A277963 G.f.: 1/(1+x) * Product_{k>=1} 1/(1-x^k)^k.

Original entry on oeis.org

1, 0, 3, 3, 10, 14, 34, 52, 108, 174, 326, 533, 946, 1539, 2628, 4251, 7046, 11288, 18313, 29017, 46261, 72533, 113942, 176841, 274353, 421680, 647065, 985593, 1497641, 2261971, 3406992, 5105317, 7628112, 11346861, 16829094, 24861952, 36623009, 53756775

Offset: 0

Vaclav Kotesovec, Nov 06 2016

nonn

Convolution of A000219 and A033999.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000219, A091360, A087787, A191659.

CoefficientList[Series[1/(1+x)*Product[1/(1-x^k)^k, {k, 1, 50}], {x, 0, 50}], x]

a(n) = Sum_{k=0..n} (-1)^(n-k)*A000219(k).

a(n) ~ Zeta(3)^(7/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * 2^(47/36) * n^(25/36)), where A = A074962 is the Glaisher-Kinkelin constant.

A291552 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) is the number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 5, 11, 25, 52, 110, 221, 444, 868, 1685, 3212, 6082, 11361, 21071, 38693, 70570, 127670, 229557, 409963, 728069, 1285522, 2258318, 3947115, 6867238, 11893648, 20513199, 35235429, 60292928, 102787903, 174620017, 295644893, 498931699, 839367287, 1407864040, 2354559426, 3926878130

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

Partial sums of A001970.

			Equivalently (Cayley), a(n) = total number of 2-dimensional partitions of all nonnegative integers <= n.
a(3) = 11 because we have:
0...1...2.11.1...3.21.2.111.11.1
.............1........1.....1..1
...............................1
and 1 + 1 + 3 + 6 = 11.

Index entries for related partition-counting sequences

Cf. A000041, A000070, A001970, A026905, A036469, A091360, A277643.

with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 11 2017

nmax = 36; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) = [x^k] Product_{k>=1} 1/(1 - x^k).

A309267 Expansion of (1 + x) * Product_{k>=1} 1/(1 - x^k)^k.

Original entry on oeis.org

1, 2, 4, 9, 19, 37, 72, 134, 246, 442, 782, 1359, 2338, 3964, 6652, 11046, 18176, 29631, 47935, 76931, 122608, 194072, 305269, 477258, 741977, 1147227, 1764778, 2701403, 4115892, 6242846, 9428575, 14181272, 21245738, 31708402, 47150928, 69867001, 103176007, 151864745, 222821779

Offset: 0

Ilya Gutkovskiy, Jul 20 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000219, A052816, A084376, A091360, A191659, A277963.

G:= (1+x)/mul((1-x^k)^k,k=1..100):
S:= series(G,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 01 2020

nmax = 38; CoefficientList[Series[(1 + x) Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[2, k] a[n - k], {k, 1, n}]/n; Table[a[n] + a[n - 1], {n, 0, 38}]

a(n) = A000219(n) + A000219(n-1).

a(n) ~ Zeta(3)^(7/36) * 2^(25/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jul 20 2019

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A096597 Triangle read by rows: T[n,m] = number of plane partitions of n whose 3-dimensional Ferrers plot just fits inside an m X m X m box, i.e., with Max[parts, rows, columns] = m.

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A277963 G.f.: 1/(1+x) * Product_{k>=1} 1/(1-x^k)^k.

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A291552 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) is the number of partitions of k (A000041).

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A309267 Expansion of (1 + x) * Product_{k>=1} 1/(1 - x^k)^k.

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