A259799 -id:A259799 - cp's OEIS frontend

A037444 Number of partitions of n^2 into squares.

1, 1, 2, 4, 8, 19, 43, 98, 220, 504, 1116, 2468, 5368, 11592, 24694, 52170, 108963, 225644, 462865, 941528, 1899244, 3801227, 7550473, 14889455, 29159061, 56722410, 109637563, 210605770, 402165159, 763549779, 1441686280, 2707535748, 5058654069, 9404116777

Offset: 0

Wouter Meeussen

Is lim_{n->inf} a(n)^(1/n) > 1? - Paul D. Hanna, Aug 20 2002

The limit above is equal to 1 (see formula by Hardy & Ramanujan for A001156). - Vaclav Kotesovec, Dec 29 2016

T. D. Noe, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..945 (terms n = 0..100 from T. D. Noe, terms n = 101..500 from Alois P. Heinz)
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

Entries with square index in A001156.
Cf. A072964, A030273, A000041, A000290, A229239, A229468.
Cf. A003108, A046042.
Cf. A259792, A259793.
A row or column of the array in A259799.

a037444 n = p (map (^ 2) [1..]) (n^2) where
   p _      0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 15 2013

max=33; se = Series[ Product[1/(1-x^(k^2)), {k, 1, max}], {x, 0, max^2}]; a[n_] := Coefficient[se, x^(n^2)]; a[0] = 1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Oct 18 2011 *)

a(n) = A001156(n^2) = coefficient of x^(n^2) in the series expansion of Prod_{k>=1} 1/(1 - x^(k^2)).

a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3)) [Hardy & Ramanujan, 1917, modified from A001156]. - Vaclav Kotesovec, Dec 29 2016

A259792 Number of partitions of n^3 into cubes.

Original entry on oeis.org

1, 1, 2, 5, 17, 62, 258, 1050, 4365, 18012, 73945, 301073, 1214876, 4852899, 19187598, 75070201, 290659230, 1113785613, 4224773811, 15866483556, 59011553910, 217410395916, 793635925091, 2871246090593, 10297627606547, 36620869115355, 129166280330900

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..173 (terms 0..120 from Alois P. Heinz)
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

A row of the array in A259799.
Cf. A279329.
Cf. A001156, A003108, A046042.
Cf. A037444, A259793.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^3>n, 0, b(n-i^3, i)))
    end:
a:= n-> b(n^3, n):
seq(a(n), n=0..26);  # Alois P. Heinz, Jul 10 2015

$RecursionLimit = 1000; b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[ i^3>n, 0, b[n-i^3, i]]]; a[n_] := b[n^3, n]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

a(n) = [x^(n^3)] Product_{j>=1} 1/(1-x^(j^3)). - Alois P. Heinz, Jul 10 2015
a(n) = A003108(n^3). - Vaclav Kotesovec, Aug 19 2015
a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * n^(3/4) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(15/4)) [after Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

More term from Alois P. Heinz, Jul 10 2015

A027601 Number of partitions of 4^n into n-th powers.

Original entry on oeis.org

5, 8, 17, 36, 88, 218, 550, 1413, 3679, 9622, 25297, 66737, 176537, 467998, 1242484, 3302290, 8784054, 23380494, 62260292, 165852148, 441922283, 1177759340, 3139292094, 8368647139, 22310795825, 59484268378, 158602342446, 422894828108, 1127630055693

Offset: 1

David W. Wilson

nonn

			a(1)=5: Partitions of 4 into natural numbers: this is A000041(4)=5.
a(2)=8: Partitions of 4^2 into squares: 16 = 16, 9 4 1^3, 9 1^7, 4^4, 4^3 1^4, 4^2 1^8, 4 1^12, 1^16. - _N. J. A. Sloane_, Jul 06 2015

H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

Cf. A000041, A027600, A027609.

A row or column of the array in A259799.

Equals A027609(n)+1

a(27)-a(29) from Alois P. Heinz, Jul 10 2015

A259793 Number of partitions of n^4 into fourth powers.

Original entry on oeis.org

1, 1, 2, 7, 36, 253, 1886, 14800, 118238, 955639, 7750456, 62777522, 506272363, 4056634991, 32252971687, 254209569990, 1985108901344, 15352968310930, 117579612410477, 891596419221856, 6694250497509934, 49768995849050468, 366423320400440927, 2671969175372760210

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..63 (terms 0..45 from Alois P. Heinz)
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

A row of the array in A259799.
Cf. A001156, A003108, A046042.
Cf. A037444, A259792.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^4>n, 0, b(n-i^4, i)))
    end:
a:= n-> b(n^4, n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 10 2015

$RecursionLimit = 10^4; b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i^4>n, 0, b[n-i^4, i]]]; a[n_] := b[n^4, n];  Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)

a(n) = [x^(n^4)] Product_{j>=1} 1/(1-x^(j^4)). - Alois P. Heinz, Jul 10 2015
a(n) = A046042(n^4). - Vaclav Kotesovec, Aug 19 2015
a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(4/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(26/5)) [after Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

More terms from Alois P. Heinz, Jul 10 2015

A259794 Number of partitions of n^5 into fifth powers.

Original entry on oeis.org

1, 1, 2, 9, 88, 1104, 15772, 241582, 3869852, 63689650, 1065023018, 17948615155, 303219868652, 5116273886322, 86004191773864, 1437703756689091, 23869446608034827, 393225674878151704, 6423761195925513669, 104014146020398166139, 1668870762057827073994

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..26
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

A row of the array in A259799.

Cf. A037444, A259792, A259793, A259795.

a(n) = [x^(n^5)] Product_{k>=1} 1/(1 - x^(k^5)). - Ilya Gutkovskiy, Jan 29 2018

More terms from Alois P. Heinz, Jul 10 2015

A259795 Number of partitions of n^6 into sixth powers.

Original entry on oeis.org

1, 1, 2, 13, 218, 5082, 140549, 4318937, 142230196, 4899174096, 173759831765, 6275666535508, 229029623773326, 8400986623582879, 308552577080828413, 11315799255444002331, 413526899811283611529, 15035521464541449037361, 543292158617220114038102, 19493012206795963934830852, 694033371089826655280448205, 24509565267500567956406898725

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

A row of the array in A259799.

Cf. A037444, A259792, A259793, A259794.

a(n) = [x^(n^6)] Product_{k>=1} 1/(1 - x^(k^6)). - Ilya Gutkovskiy, Jan 29 2018

a(8)-a(14) from Alois P. Heinz, Jul 10 2015
a(15)-a(18) from Hiroaki Yamanouchi, Jul 11 2015
a(19)-a(21) from Vaclav Kotesovec, Dec 10 2016

A259796 Number of partitions of 3^n into n-th powers.

Original entry on oeis.org

3, 4, 5, 7, 9, 13, 19, 27, 40, 59, 88, 131, 196, 293, 439, 658, 987, 1479, 2218, 3327, 4989, 7483, 11224, 16836, 25253, 37878, 56817, 85224, 127836, 191753, 287628, 431441, 647161, 970741, 1456111, 2184166, 3276248, 4914371, 7371556, 11057334, 16586000

Offset: 1

N. J. A. Sloane, Jul 06 2015

nonn

			n=3: partitions of 3^3 = 27 into cubes: 27 = 27, 8^3+1^3, 8^2+1^11, 8+1^19, 1^27, so a(3) = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

A column of the array in A259799.

More terms from Alois P. Heinz, Jul 10 2015

A331402 a(n) = [x^(n^n)] Product_{k>=1} 1 / (1 - x^(k^n)).

Original entry on oeis.org

1, 2, 5, 36, 1104, 140549, 82159688, 237700614212, 3591644060379486

Offset: 1

Ilya Gutkovskiy, Jan 16 2020

Number of partitions of n^n into n-th powers.

			a(2) = 2 because we have [4] and [1, 1, 1, 1].

Diagonal of A259799.

Cf. A145514.

a(n) = A259799(n,n).

a(8)-a(9) from Giovanni Resta, Jan 17 2020

A259797 Number of partitions of 5^n into n-th powers.

Original entry on oeis.org

7, 19, 62, 253, 1104, 5082, 24119, 117016, 577219, 2881559, 14517349, 73646606, 375647872, 1924303223, 9890611989, 50972464766, 263251147564, 1361885016990, 7055002534314, 36586652345539, 189899006940570

Offset: 1

N. J. A. Sloane, Jul 06 2015

H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

A column of the array in A259799.

a(11)-a(21) from Alois P. Heinz, Jul 10 2015

A259798 Number of partitions of 6^n into n-th powers.

Original entry on oeis.org

11, 43, 258, 1886, 15772, 140549, 1311749, 12648913, 124896943, 1255339471, 12791342074, 131747261218, 1368576060800, 14314677403795

Offset: 1

N. J. A. Sloane, Jul 06 2015

H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

A column of the array in A259799.

a(9)-a(14) from Alois P. Heinz, Jul 11 2015

cp's OEIS Frontend

A037444 Number of partitions of n^2 into squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A259792 Number of partitions of n^3 into cubes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A027601 Number of partitions of 4^n into n-th powers.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A259793 Number of partitions of n^4 into fourth powers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A259794 Number of partitions of n^5 into fifth powers.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A259795 Number of partitions of n^6 into sixth powers.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A259796 Number of partitions of 3^n into n-th powers.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A331402 a(n) = [x^(n^n)] Product_{k>=1} 1 / (1 - x^(k^n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A259797 Number of partitions of 5^n into n-th powers.

Views

Author