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A053723
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Number of 5-core partitions of n.
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8
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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; internal format)
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OFFSET
| 0,3
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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.
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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.
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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+2uvw+4uw^2-u^2w. - 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^(5k))^5/(1-x^k) = 1/x(Sum_{k>0} k*x^k(1-x^k)(1-x^(2k))/(1-x^(5k))) . - Michael Somos Jun 17 2005
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).
G.f.: x^(-1)* 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). - Michael Somos Aug 08 2007
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MATHEMATICA
| a[n_]:=Total[KroneckerSymbol[#, 5]*n/# & /@ Divisors[n]]; Table[a[n], {n, 1, 73}] (* From Jean-François Alcover, Jul 26 2011, after PARI prog. *)
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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])
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CROSSREFS
| Cf. A053724, column t=5 of A175595.
Sequence in context: A093870 A126833 * A138512 A201652 A066949 A073481
Adjacent sequences: A053720 A053721 A053722 * A053724 A053725 A053726
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KEYWORD
| easy,nonn,mult
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AUTHOR
| James A. Sellers (sellersj(AT)math.psu.edu), Feb 11 2000
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