A175595 -id:A175595 - cp's OEIS frontend

A045831 Number of 4-core partitions of n.

1, 1, 2, 3, 1, 3, 3, 3, 4, 4, 2, 2, 7, 3, 5, 6, 2, 4, 7, 3, 4, 7, 5, 8, 5, 4, 4, 8, 5, 6, 7, 2, 9, 11, 3, 8, 9, 4, 6, 5, 7, 5, 14, 7, 4, 10, 5, 10, 11, 3, 9, 10, 5, 8, 10, 4, 6, 15, 8, 9, 10, 6, 8, 15, 6, 10, 6, 5, 15, 9, 6, 8, 14, 8, 6, 13, 5, 16, 18, 7, 8, 7, 9, 6, 15, 6, 12, 17, 5, 8, 15, 7, 12

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Conjecturally  Sum_n a(n)q^(8n+5) equals theta series of sodalite. - Fred Lunnon, Mar 05 2015
Dickson writes that Liouville proved several related theorems about sums of triangular numbers. - Michael Somos, Feb 10 2020

			G.f. = 1 + x + 2*x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...
G.f. = q^5 + q^13 + 2*q^21 + 3*q^29 + q^37 + 3*q^45 + 3*q^53 + 3*q^61 + 4*q^69 + ... ,
apparently the theta series of the sodalite net, aka edge-skeleton of space honeycomb by truncated octahedra. - _Fred Lunnon_, Mar 05 2015

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. II, p. 23.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
M. Hirschhorn, and J. Sellers, Some amazing facts about 4-cores, J. Num. Thy. 60 (1996), 51-69.
K. Ono, and L. Sze, 4-core partitions and class numbers, Acta. Arith. 80 (1997), 249-272.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A004024/4, column t=4 of A175595.
Cf. A286953.
Cf. A008443, A045818, A213624.

QP = QPochhammer; s = QP[q^4]^4/QP[q] + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Jul 26 2011, updated Nov 29 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^4 / eta(x + A), n))}; /* Michael Somos, Mar 24 2003 */

eta(32*z)^4/eta(8*z) = Sum_{x, y, z} q^(x^2+2*y^2+2*z^2), x, y, z >= 1 and odd.
From Michael Somos, Mar 24 2003: (Start)
Euler transform of period 4 sequence [1, 1, 1, -3, ...].
Expansion of q^(-5/8) * eta(q^4)^4/eta(q) in powers of q.
(End)
Number of solutions to n=t1+2*t2+2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006
G.f.: Product_{k>0} (1-q^(4*k))^4/(1-q^k).
Expansion of psi(q) * psi(q^2)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Sep 02 2008

More terms from James Sellers, Feb 11 2000

A053723 Number of 5-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 6, 5, 7, 5, 12, 6, 12, 6, 10, 11, 16, 7, 20, 15, 12, 12, 22, 10, 25, 12, 20, 18, 30, 10, 32, 21, 24, 16, 30, 21, 36, 20, 24, 25, 42, 12, 42, 36, 35, 22, 46, 22, 43, 25, 32, 36, 52, 20, 60, 30, 40, 30, 60, 30, 62, 32, 42, 43, 60, 24, 66, 48, 44, 30, 72, 35, 72

Offset: 0

James Sellers, Feb 11 2000

Number 11 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 5*x^7 + 7*x^8 + ...
G.f. = q + q^2 + 2*q^3 + 3*q^4 + 5*q^5 + 2*q^6 + 6*q^7 + 5*q^8 + 7*q^9 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see p. 54 (1.52).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Subhajit Bandyopadhyay and Nayandeep Deka Baruah, Arithmetic Identities for Some Analogs of the 5-Core Partition Function, J. Int. Seq. (2024) Vol. 27, Art. No. 24.4.5.
Subhajit Bandyopadhyay and Nayandeep Deka Baruah, Arithmetic Identities for Some Analogs of 5-core Partition Function, arXiv:2409.02034 [math.NT], 2024.
Bruce C. Berndt, The Rogers-Ramanujan continued fraction.
Bruce C. Berndt et al., The Rogers-Ramanujan continued fraction, Journal of Computational and Applied Mathematics, Vol. 105, No. 1-2 (1999), 9-24.
Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652. see pages 636-637.
Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and t-cores.
Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Column t=5 of A175595.
Cf. A053724, A109063, A109064, A138512.
Cf. A227216, A229802, A328717.

a[n_]:=Total[KroneckerSymbol[#, 5]*n/# & /@ Divisors[n]]; Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Jul 26 2011, after PARI prog. *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^5]^5 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jul 13 2012 *)
a[ n_] := With[{m = n + 1}, If[ m < 1, 0, DivisorSum[ m, m/# KroneckerSymbol[ 5, #] &]]]; (* Michael Somos, Jul 13 2012 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A)^5 / eta(x + A), n))};

{a(n) = if( n<0, 0, n++; sumdiv( n, d, kronecker( d, 5) * n/d))};

{a(n) = if( n<0, 0, n++; direuler( p=2, n, 1 / ((1 - p*X) * (1 - kronecker( p, 5) * X)))[n])};

Given g.f. A(x), then B(q) = q * A(q) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 2 * u*v*w + 4 * u*w^2 - u^2*w. - Michael Somos, May 02 2005
G.f.: (1/x) * (Sum_{k>0} Kronecker(k, 5) * x^k / (1 - x^k)^2). - Michael Somos, Sep 02 2005
G.f.: Product_{k>0} (1 - x^(5*k))^5 / (1 - x^k) = 1/x * (Sum_{k>0} k * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k))). - Michael Somos, Jun 17 2005
G.f.: (1/x) * Sum_{a, b, c, d, e in Z^5} x^((a^2 + b^2 + c^2 + d^2 + e^2) / 10) where a + b + c + d + e = 0, (a, b, c, d, e) == (0, 1, 2, 3, 4) (mod 5). - [Dyson 1972] Michael Somos, Aug 08 2007
Euler transform of period 5 sequence [ 1, 1, 1, 1, -4, ...].
Expansion of q^(-1) * eta(q^5)^5 / eta(q) in powers of q.
a(n) = b(n + 1) where b() is multiplicative with b(5^e) = 5^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), b(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).
Convolution inverse of A109063. a(n) = (-1)^n * A138512(n+1).
Convolution of A227216 and A229802. - Michael Somos, Jun 10 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = (1/5)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109064. - Michael Somos, May 17 2015
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328717. - Amiram Eldar, Nov 23 2023

A053724 Number of 7-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 8, 15, 16, 21, 21, 28, 24, 44, 36, 49, 45, 63, 49, 74, 64, 85, 72, 105, 82, 133, 112, 120, 120, 165, 122, 180, 147, 186, 176, 225, 168, 255, 210, 245, 224, 324, 219, 338, 276, 341, 294, 385, 288, 441, 352, 410, 366, 518, 360, 506, 435, 504

Offset: 0

James Sellers, Feb 11 2000

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 8*x^7 + 15*x^8 + ...
G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 8*q^9 + ...

A. Balog, H. Darmon, K. Ono, Congruence for Fourier coefficients of half-integral weight modular forms and special values of L-functions, pp. 105-128 of Analytic number theory, Vol. 1, Birkhauser, Boston, 1996, see page 107.
B. Berndt, Commentary on Ramanujan's Papers, pp. 357-426 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 372 (4).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
A. Berkovich and H. Yesilyurt, New identities for 7-cores with prescribed BG-rank, Discrete Math., 308 (2008), 5246-5259.
F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17,
B. Kim, On inequalities and linear relations for 7-core partitions, Discrete Math., 310 (2010), 861-868.
K. Saito, Eta-produkt eta(7tau)^7/eta(tau), arXiv:math/0602367 [math.NT], 2006.

Cf. A053723, column t=7 of A175595.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^7]^7 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^7 + A)^7 / eta(x + A), n))}; /* Michael Somos, Apr 16 2005 */

Expansion of q^(-2) * eta(q^7)^7 / eta(q) in powers of q.
Euler transform of period 7 sequence [ 1, 1, 1, 1, 1, 1, -6, ...].
a(7*n + 5) == 0 (mod 7).
G.f.: Product_{k>0} (1 - q^(7*k))^7 / (1 - q^k).

A286950 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - x^(k*j))^k.

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, -1, 0, 0, 1, -1, 1, 0, 0, 1, -1, -1, -2, 0, 1, 1, -1, -1, 3, 3, 0, 0, 1, -1, -1, 0, -3, -4, 0, 1, 1, -1, -1, 0, 4, -2, 5, 0, 0, 1, -1, -1, 0, 0, -3, 9, -7, 0, 0, 1, -1, -1, 0, 0, 6, -4, -8, 10, 0, 0, 1, -1, -1, 0, 0, 1, -5, 1, -6, -13, 0, 0, 1

Offset: 0

Seiichi Manyama, May 17 2017

			Square array begins:
   1, 1,  1,  1,  1, ...
  -1, 0, -1, -1, -1, ...
  -1, 0,  1, -1, -1, ...
   0, 0, -2,  3,  0, ...
   0, 0,  3, -3,  4, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A010815, A000007, A106507, A286952, A286953.
Diagonal gives A286956.
Cf. A175595.

G.f. of column k: Product_{j>=1} (1 - x^j)/(1 - x^(k*j))^k.

A182803 Number of 8-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 14, 22, 26, 32, 37, 45, 47, 56, 75, 77, 89, 102, 111, 124, 142, 147, 167, 182, 196, 210, 242, 249, 288, 322, 299, 349, 382, 393, 423, 467, 453, 499, 570, 563, 602, 669, 649, 716, 772, 754, 843, 907, 884

Offset: 0

Alois P. Heinz, Dec 03 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

8th column of A175595.

with(numtheory): A:= proc(n, t) option remember; local d, j; `if`(n=0, 1, add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d), d=divisors(j)) *A(n-j, t), j=1..n)/n) end: seq(A(n,8), n=0..50);

A[n_, t_] := A[n, t] = If[n == 0, 1, Sum[Sum[If[t == 0 || Mod[d, t] == 0, d - d*t, d], {d, Divisors[j]}]*A[n - j, t], {j, 1, n}]/n];
Table[A[n, 8], {n, 0, 50}] (* Jean-François Alcover, May 21 2018, translated form Maple *)

G.f.: Product_{i>=1} (1-x^(8*i))^8/(1-x^i).

Euler transform of period 8 sequence [1,1,1,1,1,1,1,-7, .. ].

A182804 Number of 9-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 21, 33, 38, 50, 56, 72, 77, 96, 99, 142, 139, 177, 180, 228, 229, 288, 284, 357, 343, 430, 410, 519, 491, 615, 588, 745, 714, 832, 811, 1007, 939, 1152, 1077, 1310, 1215, 1456, 1426, 1686, 1580, 1887, 1778, 2137

Offset: 0

Alois P. Heinz, Dec 03 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

9th column of A175595.

with(numtheory): A:= proc(n, t) option remember; local d, j; `if`(n=0, 1, add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d), d=divisors(j)) *A(n-j, t), j=1..n)/n) end: seq(A(n,9), n=0..50);

A[n_, t_] := A[n, t] = If[n == 0, 1, Sum[Sum[If[t == 0 || Mod[d, t] == 0, d - d*t, d], {d, Divisors[j]}]*A[n - j, t], {j, 1, n}]/n] ;
Table[A[n, 9], {n, 0, 50}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

G.f.: Product_{i>=1} (1-x^(9*i))^9/(1-x^i).

Euler transform of period 9 sequence [1,1,1,1,1,1,1,1,-8, .. ].

A182805 Number of 10-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 32, 46, 57, 71, 85, 106, 121, 147, 165, 190, 242, 267, 302, 350, 400, 443, 511, 565, 638, 715, 774, 852, 964, 1038, 1135, 1253, 1372, 1482, 1650, 1785, 1878, 2098, 2234, 2411, 2625, 2819, 2963, 3249, 3393, 3600, 4004, 4181

Offset: 0

Alois P. Heinz, Dec 03 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

10th column of A175595.

Cf. A033687, A045831, A053723, A081622, A053724, A182803, A182804.

with(numtheory): A:= proc(n, t) option remember; local d, j; `if`(n=0, 1, add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d), d=divisors(j)) *A(n-j, t), j=1..n)/n) end: seq(A(n,10), n=0..50);

A[n_, t_] := A[n, t] = Module[{d, j}, If[n == 0, 1, Sum[Sum[If[t == 0 || Mod[d, t] == 0, d - d t, d], {d, Divisors[j]}] A[n - j, t], {j, 1, n}]/n]];
Table[A[n, 10], {n, 0, 50}] (* Jean-François Alcover, Dec 06 2020, after Alois P. Heinz *)

G.f.: Product_{i>=1} (1-x^(10*i))^10/(1-x^i).

Euler transform of period 10 sequence [1,1,1,1,1,1,1,1,1,-9, .. ].

A363675 Numbers k such that the least common multiple of the degrees of the irreducible characters of S_k equals |S_k| = k!.

Original entry on oeis.org

0, 1, 6, 10, 21, 36, 66, 105, 120, 136, 190, 210, 276, 325, 465, 496, 561, 630, 666, 741, 780, 990, 1081, 1176, 1225, 1540, 1596, 1830, 2080, 2145, 2346, 2556, 2926, 3081, 3160, 3240, 3486, 3570, 3916, 4005, 4186, 4560, 4656, 4950, 5050, 5356, 5460, 5886, 6105

Offset: 1

Diego Martin Duro, Jun 14 2023

nonn

Intersection of the sequences of numbers k such that there exists a 2-core partition of k (A267137) and a 3-core partition of k (A000217).

Alois P. Heinz, Table of n, a(n) for n = 1..100
Diego Martin Duro, Arithmetic Properties of Character Degrees and the Generalised Knutson Index, arXiv:2306.12408 [math.RT], 2023.

Cf. A000142, A000217, A010054, A033687, A175595, A260682, A267137, A363676, A363701.

From Alois P. Heinz, Jun 16 2023: (Start)
{ k : A175595(k,2) > 0 and A175595(k,3) > 0 }.
{ k : A010054(k) > 0 and A033687(k) > 0 }. (End)

More terms from Alois P. Heinz, Jun 16 2023

A053691 Number of 11-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 45, 66, 79, 102, 121, 154, 176, 220, 248, 297, 330, 430, 452, 552, 605, 720, 777, 935, 990, 1182, 1265, 1485, 1530, 1838, 1892, 2214, 2310, 2684, 2750, 3238, 3289, 3850, 3960, 4500, 4599, 5370, 5404, 6220, 6325, 7238

Offset: 0

James Sellers, Feb 14 2000

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = q^5 + q^6 + 2*q^7 + 3*q^8 + 5*q^9 + 7*q^10 + 11*q^11 + 15*q^12 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Column t=11 of A175595.

m = 50; CoefficientList[ Series[ Product[(1-q^(11*k))^11/(1-q^k), {k, 1, m}], {q, 0, m}], q] (* Jean-François Alcover, Jul 26 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^11]^11 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 06 2014 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^11 + A)^11 / eta(x + A), n))}; /* Michael Somos, Nov 06 2014 */

Expansion of f(-x^11)^11 / f(-x) in powers of x where f() is a Ramanujan theta function.
Expansion of q^-5 * etq(q^11)^11 / eta(q) in powers of q. - Michael Somos, Nov 06 2014
Euler transform of period 11 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -10, ...]. - Michael Somos, Nov 06 2014
G.f. Product_{k>0} (1 - x^(11*k))^11 / (1 - x^k).

A192061 Number of 12-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 65, 89, 111, 140, 171, 213, 253, 310, 363, 432, 498, 583, 705, 800, 924, 1060, 1216, 1379, 1578, 1772, 2013, 2259, 2554, 2847, 3147, 3507, 3897, 4305, 4756, 5225, 5748, 6297, 6909, 7546, 8250, 9000, 9724, 10626, 11512, 12478, 13482, 14616, 15714, 17007, 18215, 19602, 20930, 22470

Offset: 0

N. J. A. Sloane, Jun 21 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

A column of A175595.

M:=60;
f:=proc(t) global M; local q,i,t1;
t1:=1;
for i from 1 to M+1 do
t1:=series(t1*(1-q^(i*t))^t,q,M);
t1:=series(t1/(1-q^i),q,M);
od;
t1;
end;
seriestolist(f(12));

m = 60;
f[0] = 1/(1 - x^k);
f[t_] = (1 - x^(k t))^t/(1 - x^k);
s[t_] := CoefficientList[Series[Product[f[t], {k, m-1}], {x, 0, m-1}], x];
s[12] (* Jean-François Alcover, May 27 2020 *)

cp's OEIS Frontend

A045831 Number of 4-core partitions of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A053723 Number of 5-core partitions of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A053724 Number of 7-core partitions of n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A286950 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - x^(k*j))^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A182803 Number of 8-core partitions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A182804 Number of 9-core partitions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A182805 Number of 10-core partitions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula