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A053723 Number of 5-core partitions of n. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

B. C. Berndt, H. H. Chan, S.-S. Huang, S.-Y. Kang, J. Sohn and S. H. Son, The Rogers-Ramanujan continued fraction, J. Comput. Appl. Math. 105 (1999), 9-24.

Garvan, F., Kim, D. and Stanton, D., Cranks and t-cores, Inventiones Math. 101 (1990), 1-17.

Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652. see pages 636-637.

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see p. 54 (1.52).

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

B. C. Berndt, The Rogers-Ramanujan continued fraction.

F. Garvan, D. Kim and D. Stanton, Cranks and t-cores.

FORMULA

Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2), B(x^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} 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(n) 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).

EXAMPLE

1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 5*x^7 + 7*x^8 + ...

q + q^2 + 2*q^3 + 3*q^4 + 5*q^5 + 2*q^6 + 6*q^7 + 5*q^8 + 7*q^9 + ...

MATHEMATICA

a[n_]:=Total[KroneckerSymbol[#, 5]*n/# & /@ Divisors[n]]; Table[a[n], {n, 1, 73}] (* Jean-Fran├žois Alcover, Jul 26 2011, after PARI prog. *)

a[ n_] := SeriesCoefficient[ QPochhammer[ q^5]^5 / QPochhammer[ q], {q, 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 *)

PROG

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

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

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

CROSSREFS

Cf. A053724, A109063, A138512. column t=5 of A175595.

Sequence in context: A239692 A126833 * A138512 A201652 A066949 A073481

Adjacent sequences:  A053720 A053721 A053722 * A053724 A053725 A053726

KEYWORD

easy,nonn,mult

AUTHOR

James A. Sellers, Feb 11 2000

STATUS

approved

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Last modified April 19 17:27 EDT 2014. Contains 240767 sequences.