A053724 -id:A053724 - cp's OEIS frontend

A053723 Number of 5-core partitions of n.

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see p. 54 (1.52).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

A175595 Square array A(n,t), n>=0, t>=0, read by antidiagonals: A(n,t) is the number of t-core partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 1, 0, 0, 5, 1, 1, 2, 1, 0, 7, 1, 1, 2, 0, 0, 0, 11, 1, 1, 2, 3, 2, 0, 0, 15, 1, 1, 2, 3, 1, 1, 1, 0, 22, 1, 1, 2, 3, 5, 3, 2, 0, 0, 30, 1, 1, 2, 3, 5, 2, 3, 0, 0, 0, 42, 1, 1, 2, 3, 5, 7, 6, 3, 1, 0, 0, 56, 1, 1, 2, 3, 5, 7, 5, 5, 4, 2, 1, 0, 77, 1, 1, 2, 3, 5, 7, 11, 9, 7, 4, 2, 0, 0, 101

Offset: 0

Alois P. Heinz, Dec 03 2010

A partition of n is a t-core partition if none of the hook numbers associated to the Ferrers-Young diagram is a multiple of t. See Chen reference for definitions.

			A(4,3) = 2, because there are 2 partitions of 4 such that no hook number is a multiple of 3:
   (1)  2 | 4 1
       +1 | 2
       +1 | 1
   -------+-----
   (2)  3 | 4 2 1
       +1 | 1
Square array A(n,t) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  ...
   1,  0,  1,  1,  1,  1,  1,  1,  ...
   2,  0,  0,  2,  2,  2,  2,  2,  ...
   3,  0,  1,  0,  3,  3,  3,  3,  ...
   5,  0,  0,  2,  1,  5,  5,  5,  ...
   7,  0,  0,  1,  3,  2,  7,  7,  ...
  11,  0,  1,  2,  3,  6,  5, 11,  ...
  15,  0,  0,  0,  3,  5,  9,  8,  ...

Garvan, F. G., A number-theoretic crank associated with open bosonic strings. In Number Theory and Cryptography (Sydney, 1989), 221-226, London Math. Soc. Lecture Note Ser., 154, Cambridge Univ. Press, Cambridge, 1990.
James, Gordon; and Kerber, Adalbert, The Representation Theory of the Symmetric Group. Addison-Wesley Publishing Co., Reading, Mass., 1981.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, arXiv:math/0208050 [math.NT], 2002.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, Rankin memorial issues. Ramanujan J. 7 (2003), 343-366.
Shichao Chen, Arithmetical properties of the number of t-core partitions, The Ramanujan Journal, 18 (2007), no. 1, 103-112, DOI: 10.1007/s11139-007-9045-5.
F. G. Garvan, The crank of partitions mod 8, 9 and 10, Trans. Amer. Math. Soc. 322 (1990), 79-94.
F. G. Garvan, Some congruences for partitions that are p-cores, Proc. London Math. Soc. 66 (1993), 449-478.
F. G. Garvan, More cranks and t-cores, Bull. Austral. Math. Soc. 63 (2001), 379-391.
F. G. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
Ben Kane, Sums of Triangular Numbers and t-Core Partitions, Journal of Combinatorics and Number Theory, 1 (2009), no.1, 59-64.
B. Kim, On inequalities and linear relations for 7-core partitions, Discrete Math., 310 (2010), 861-868.
N. J. A. Sloane, Transforms.

Columns t=0-12 give A000041, A000007, A010054, A033687, A045831, A053723, A081622, A053724, A182803, A182804, A182805, A053691, A192061.
Rows n=0-1 give A000012, A060576.
Diagonal gives A000094(n+1) for n>0.
Upper diagonal gives A000041.
Lower diagonal (conjectured) gives A086642 for n>0.

with(numtheory):
A:= proc(n, t) option remember; `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(seq(A(n, d-n), n=0..d), d=0..14);
(From N. J. A. Sloane, Jun 21 2011: to get M terms of the series for t-core partitions:)
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;
# then for example seriestolist(f(5));

n = 13; f[t_] = (1-x^(t*k))^t/(1-x^k); f[0] = 1/(1-x^k);
s[t_] := CoefficientList[ Series[ Product[ f[t], {k, 1, n}], {x, 0, n}], x]; m = Table[ PadRight[ s[t], n+1], {t, 0, n}]; Flatten[ Table[ m[[j+1-k, k]], {j, n+1}, {k, j}]] (* Jean-François Alcover, Jul 25 2011, after g.f. *)

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

Column t is the Euler transform of period t sequence [1, .., 1, 1-t, ..].

Additional references from N. J. A. Sloane, Jun 21 2011

A081622 Number of 6-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 5, 9, 10, 12, 12, 14, 20, 20, 21, 23, 24, 24, 32, 29, 35, 36, 44, 47, 38, 47, 49, 52, 55, 58, 59, 64, 66, 71, 70, 78, 79, 88, 87, 90, 85, 87, 111, 104, 102, 107, 112, 113, 121, 113, 130, 130, 148, 153, 132, 147, 149, 156, 162, 149, 167, 160, 178, 180

Offset: 0

Michael Somos, Mar 24 2003

Euler transform of period 6 sequence [ 1, 1, 1, 1, 1, -5, ...].

Expansion of q^(-35/24) * eta(q^6)^6 / eta(q) in powers of q.

			1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 5*x^6 + 9*x^7 + 10*x^8 + 12*x^9 + ...
q^35 + q^59 + 2*q^83 + 3*q^107 + 5*q^131 + 7*q^155 + 5*q^179 + 9*q^203 + ...

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.

Cf. A010054, A033687, A045831, A053723, A053724.

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^(6*k) + x * O(x^n))^6 / (1 - x^k)), n))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^6 / eta(x + A), n))}

G.f.: Product_{k>0} (1 - x^(6*k))^6 / (1 - x^k).

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, .. ].

A194965 Fractalization of (A053824(n+5)), n>=0.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 6, 2, 3, 4, 5, 1, 6, 7, 2, 3, 4, 5, 1, 6, 7, 8, 2, 3, 4, 5, 1, 6, 7, 8, 9, 2, 3, 4, 5, 1, 6, 7, 8, 9, 10, 2, 3, 4, 5, 1, 6, 11, 7, 8, 9, 10, 2, 3, 4, 5, 1, 6, 11, 12, 7, 8, 9, 10, 2, 3, 4, 5, 1, 6, 11, 12, 13, 7, 8, 9, 10, 2, 3, 4, 5, 1, 6, 11

Offset: 1

Clark Kimberling, Sep 07 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (A053724(n+5)), n>=0 is formed by concatenating 5-tuples of the form (n,n+1,n+2, n+3,n+4) for n>=1: 1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,...

Cf. A194959, A194965, A194966, A194967.

p[n_] := Floor[(n + 4)/5] + Mod[n - 1, 5]
Table[p[n], {n, 1, 90}]  (* A053824(n+5), n>=0 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]   (* A194965 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194966 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A194967 *)

A105634 Expansion of Sum_{k>0} Kronecker(k,7)x^k(1 + x^k)/(1 - x^k)^3.

Original entry on oeis.org

1, 5, 8, 21, 24, 40, 49, 85, 73, 120, 122, 168, 168, 245, 192, 341, 288, 365, 360, 504, 392, 610, 530, 680, 601, 840, 656, 1029, 842, 960, 960, 1365, 976, 1440, 1176, 1533, 1370, 1800, 1344, 2040, 1680, 1960, 1850, 2562, 1752, 2650, 2208, 2728, 2401, 3005

Offset: 1

Michael Somos, Apr 16 2005, Mar 31 2008

			q + 5*q^2 + 8*q^3 + 21*q^4 + 24*q^5 + 40*q^6 + 49*q^7 + 85*q^8 + 73*q^9 + ...

A. Balog, H. Darmon and 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, Birkhäuser, Boston, 1996, see page 107.
Bruce 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).

Cf. A002656, A053724, A138809, A327135.

f[p_, e_] := If[MemberQ[{1, 2, 4}, Mod[p, 7]], (p^(2*e+2)-1)/(p^2-1), (p^(2*e+2)+(-1)^e)/(p^2+1)]; f[7, e_] := 7^(2*e); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)

{a(n)=local(A,p,e); if(n<2, n==1, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==7, p^(2*e), if(kronecker(p,7)==1, (p^(2*e+2)-1)/(p^2-1), (p^(2*e+2)+(-1)^e)/(p^2+1)))))) }

{a(n)=local(A,B); if(n<1, 0, n--; A=x*O(x^n); polcoeff( if(B=eta(x^7+A), A=eta(x+A); (A*B)^3+8*x*B^7/A), n))}

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-7, n / d)))}

Multiplicative with a(p^e) = p^(2e) if p = 7; (p^(2e+2)-1)/(p^2-1) if p == 1, 2, 4 (mod 7); (p^(2e+2)+(-1)^e)/(p^2+1) if p == 3, 5, 6 (mod 7).
G.f.: Sum_{k>0} Kronecker(k, 7)*x^k*(1+x^k)/(1-x^k)^3.
a(n) = A002656(n) + 8*A053724(n-2).
a(7n) = 49a(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(-1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A138809.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 32*Pi^3/(343*sqrt(7)) = 1.093343069... (A327135). - Amiram Eldar, Nov 16 2023

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A053723 Number of 5-core partitions of n.

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A175595 Square array A(n,t), n>=0, t>=0, read by antidiagonals: A(n,t) is the number of t-core partitions of n.

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A081622 Number of 6-core partitions of n.

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A182805 Number of 10-core partitions of n.

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A194965 Fractalization of (A053824(n+5)), n>=0.

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A105634 Expansion of Sum_{k>0} Kronecker(k,7)*x^k*(1 + x^k)/(1 - x^k)^3.

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A105634 Expansion of Sum_{k>0} Kronecker(k,7)x^k(1 + x^k)/(1 - x^k)^3.