A007327 -id:A007327 - cp's OEIS frontend

A007326 Difference between A000294 and the number of solid partitions of n (A000293).

0, 0, 0, 0, 0, 0, 1, 3, 8, 19, 40, 83, 176, 365, 775, 1643, 3483, 7299, 15170, 31010, 62563, 124221, 243296, 469856, 896491, 1690475, 3155551, 5834871, 10701036, 19479021, 35227889, 63335778, 113286272, 201687929, 357585904, 631574315, 1111614614, 1950096758, 3410420973, 5946337698, 10337420278, 17918573379, 30968896662, 53366449357, 91689380979, 157058043025, 268210414468, 456613323892

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

Understanding this sequence is a famous unsolved problem in the theory of partitions.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..72
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

a(n) = A000294(n) - A000293(n).

Cf. A007327, A007328, A007329, A007330, A008780, A042984.

Entry revised by Sean A. Irvine and N. J. A. Sloane, Dec 18 2017

A008780 a(n) = (n-dimensional partitions of 6) + C(n,4) + C(n,3).

Original entry on oeis.org

1, 11, 48, 141, 331, 672, 1232, 2094, 3357, 5137, 7568, 10803, 15015, 20398, 27168, 35564, 45849, 58311, 73264, 91049, 112035, 136620, 165232, 198330, 236405, 279981, 329616, 385903, 449471, 520986, 601152, 690712, 790449, 901187, 1023792, 1159173, 1308283

Offset: 0

N. J. A. Sloane

These are the conjectured numbers of d-dimensional partitions for n=6, coming from a formula proposed by MacMahon in the general case that turned out to be wrong. Still, here for n=6, MacMahon's formula coincides for d < 3 with the first three terms of A042984. - Michel Marcus, Aug 16 2013

Binomial transform of [1,10,27,29,12,1,0,0,0,...], 6th row of A116672. - R. J. Mathar, Jul 18 2017

G. E. Andrews, The Theory of Partitions, Add.-Wes. '76, p. 190.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A007326, A007327, A042984, A116672.

List([0..40], n-> (120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120); # G. C. Greubel, Sep 11 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+5*x-3*x^2-2*x^3)/(1-x)^6 )); // G. C. Greubel, Sep 11 2019

seq(1+10*n+27*binomial(n,2)+29*binomial(n,3)+12*binomial(n,4)+binomial(n,5), n=0..40);

Table[1+10n+27Binomial[n,2]+29Binomial[n,3]+12Binomial[n,4]+ Binomial[n,5], {n,0,40}] (* Harvey P. Dale, Jul 27 2011 *)
CoefficientList[Series[(1+5x-3x^2-2x^3)/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 17 2013 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,48,141,331,672},40] (* Harvey P. Dale, Aug 28 2019 *)

my(x='x+O('x^40)); Vec((1+5*x-3*x^2-2*x^3)/(1-x)^6) \\ G. C. Greubel, Sep 11 2019

[(120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120 for n in (0..40)] # G. C. Greubel, Sep 11 2019

G.f.: (1 + 5*x - 3*x^2 - 2*x^3)/(1-x)^6. - Colin Barker, Sep 05 2012
From G. C. Greubel, Sep 11 2019: (Start)
a(n) = (120 + 404*n + 490*n^2 + 255*n^3 + 50*n^4 + n^5)/120.
E.g.f.: (120 + 1200*x + 1620*x^2 + 580*x^3 + 60*x^4 + x^5)*exp(x)/120. (End)

Description corrected by Alford Arnold, Aug 1998

More terms added by G. C. Greubel, Sep 11 2019

A042984 Number of n-dimensional partitions of 6.

Original entry on oeis.org

1, 11, 48, 140, 326, 657, 1197, 2024, 3231, 4927, 7238, 10308, 14300, 19397, 25803, 33744, 43469, 55251, 69388, 86204, 106050, 129305, 156377, 187704, 223755, 265031, 312066, 365428, 425720, 493581, 569687, 654752, 749529, 854811, 971432

Offset: 0

Alford Arnold, Aug 15 1998

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A007326, A007327, A008780.

List([0..40],n->(n+1)*(n+4)*(n^3+40*n^2+61*n+30)/120); # Muniru A Asiru, Feb 17 2019

[1 + 10*n + 27*Binomial(n,2) + 28*Binomial(n,3) + 11*Binomial(n,4) + Binomial(n,5): n in [0..40]]; // Vincenzo Librandi, Oct 27 2013

a:= n-> 1+10*n+27*binomial(n, 2)+28*binomial(n, 3)
              +11*binomial(n, 4)+binomial(n, 5):
seq(a(n), n=0..34);

LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,48,140,326,657},40] (* Harvey P. Dale, Jan 27 2013 *)
CoefficientList[Series[(x^4 -3x^3 -3x^2 +5x +1)/(x-1)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 27 2013 *)

my(x='x+O('x^40)); Vec((x^4-3*x^3-3*x^2+5*x+1)/(x-1)^6) \\ G. C. Greubel, Feb 17 2019

((x^4-3*x^3-3*x^2+5*x+1)/(x-1)^6).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 17 2019

a(n) = A008780(n) - binomial(n, 4) - binomial(n, 3).
G.f.: (x^4 - 3*x^3 - 3*x^2 + 5*x + 1)/(x-1)^6. - Colin Barker, Jul 22 2012
a(n) = (n+1)*(n+4)*(n^3 + 40*n^2 + 61*n + 30)/120. - Robert Israel, Jul 06 2016

More terms from Erich Friedman

A007328 Difference between the number of 5-dimensional partitions of n and an approximation derived from binomial(n,4).

Original entry on oeis.org

0, 0, 0, 0, 0, 15, 75, 310, 1060, 3281, 9564, 26719, 72239, 191569, 500797, 1299925, 3362473, 8697198, 22513878, 58352126, 151267141, 391728632, 1011734975, 2602330120, 6657204192, 16920629023, 42697311397, 106912113623, 265560809521, 654270114555

Offset: 1

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100; alternative link.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Cf. A000390, A000391.

Cf. A007326, A007327, A007329, A007330.

a(n) = A000391(n) - A000390(n). - Sean A. Irvine, Dec 18 2017

a(11)-a(21) from Sean A. Irvine, Dec 18 2017

More terms from Amiram Eldar, May 11 2024

A007329 Unexplained difference between two partition generating functions.

Original entry on oeis.org

0, 0, 0, 0, 0, 35, 210, 1001, 3927, 13971, 46592, 148337, 455609, 1362656, 3989914, 11504669, 32804967, 92877609, 261846522

Offset: 1

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Cf. A000416, A000417.

Cf. A007326, A007327, A007328, A007330.

a(n) = A000417(n) - A000416(n). - Sean A. Irvine, Dec 18 2017

a(11)-a(19) from Sean A. Irvine, Dec 18 2017

A007330 Difference between the number of 7-dimensional partitions of n and an approximation derived from binomial(n,6).

Original entry on oeis.org

0, 0, 0, 0, 0, 70, 490, 2632, 11606, 46375, 173362, 618086, 2123709, 7086864, 23085942, 73761644, 232002909, 720819622, 2218608551, 6782480955

Offset: 1

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100; alternative link.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Cf. A000427, A000428, A007326, A007327, A007328, A007329.

a(n) = A000428(n) - A000427(n). - Sean A. Irvine, Dec 18 2017

a(11)-a(18) from Sean A. Irvine, Dec 18 2017

a(19)-a(20) from Amiram Eldar, May 11 2024

cp's OEIS Frontend

A007326 Difference between A000294 and the number of solid partitions of n (A000293).

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A008780 a(n) = (n-dimensional partitions of 6) + C(n,4) + C(n,3).

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A042984 Number of n-dimensional partitions of 6.

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A007328 Difference between the number of 5-dimensional partitions of n and an approximation derived from binomial(n,4).

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A007329 Unexplained difference between two partition generating functions.

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A007330 Difference between the number of 7-dimensional partitions of n and an approximation derived from binomial(n,6).

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