A008780 -id:A008780 - 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

A007327 Difference between two partition g.f.s.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 20, 69, 200, 521, 1294, 3126, 7364, 17309, 40577, 95460, 224971, 531368, 1252664, 2943095, 6870029, 15911618, 36507381, 82930347, 186414619, 414654766, 912766795, 1989007381, 4292038414, 9175624264, 19442250125, 40851448761, 85157787033, 176200110937

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

George 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).

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. A000334, A000335.

Cf. A007326, A007328, A007329, A007330, A008780, A042984.

a(n) = A000335(n) - A000334(n). - Sean A. Irvine, Dec 18 2017

a(11)-a(23) from Sean A. Irvine, Dec 18 2017

More terms from Amiram Eldar, May 11 2024

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

A116672 Triangle read by rows in which the binomial transform of the n-th row gives the Euler transform of the n-th diagonal of Pascal's triangle (A007318).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 7, 1, 1, 10, 27, 29, 12, 1, 1, 14, 57, 96, 72, 21, 1, 1, 21, 117, 277, 319, 176, 38, 1

Offset: 1

Alford Arnold, Feb 22 2006

For example, the Euler transform of 1,3,6,... is 1,1,4,10,26,59,141,... (A000294) differing slightly from A000293 which counts the solid partitions.
The NAME does not reproduce the DATA, COMMENTS, or EXAMPLES.  - R. J. Mathar, Jul 19 2017
The binomial transforms of the rows form the rows of A289656. - N. J. A. Sloane, Jul 19 2017

			Row 6 is 1 10 27 29 12 1 generating 1 11 48 141 ... (A008780) the seventh term in the Euler transforms of 1,1,1,...; 1,2,3,...; 1,3,6,... 1,4,10,... etc.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 6, 11, 7, 1;
1, 10, 27, 29, 12, 1;
1, 14, 57, 96, 72, 21, 1;
1, 21, 117, 277, 319, 176, 38, 1;
...

N. J. A. Sloane, Transforms

Cf. A000293, A116673 (row sums), A008778 - A008780, A289656.

A289656 Triangle read by rows: row n gives the first n terms of the binomial transform of the n-th row of A116672.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 5, 13, 26, 1, 7, 24, 59, 120, 1, 11, 48, 141, 331, 672, 1, 15, 86, 310, 855, 1982, 4067, 1, 22, 160, 692, 2214, 5817, 13301, 27428

Offset: 1

N. J. A. Sloane, Jul 19 2017

Rows 4, 5, 6 match the starts of sequences A008778, A008779, A008780.

			Triangle begins:
[1]
[1, 2]
[1, 3, 6]
[1, 5, 13, 26]
[1, 7, 24, 59, 120]
[1, 11, 48, 141, 331, 672]
[1, 15, 86, 310, 855, 1982, 4067]
[1, 22, 160, 692, 2214, 5817, 13301, 27428]
...

N. J. A. Sloane, Transforms

Cf. A116672, A008778, A008779, A008780.

cp's OEIS Frontend

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

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A007327 Difference between two partition g.f.s.

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A042984 Number of n-dimensional partitions of 6.

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A116672 Triangle read by rows in which the binomial transform of the n-th row gives the Euler transform of the n-th diagonal of Pascal's triangle (A007318).

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A289656 Triangle read by rows: row n gives the first n terms of the binomial transform of the n-th row of A116672.

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