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A103263
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Number of partitions of n into distinct parts prime to 3 and 5.
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1
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1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 18, 20, 22, 25, 28, 30, 33, 36, 39, 43, 48, 52, 56, 61, 67, 73, 80, 87, 94, 101, 110, 120, 130, 141, 152, 164, 177, 192, 207, 223, 240, 258, 278, 301, 324, 348, 373, 400, 429, 461, 496
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OFFSET
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0,8
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LINKS
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FORMULA
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Expansion of q^(-1/3)(eta(q^2)*eta(q^3)*eta(q^5)*eta(q^30))/(eta(q)*eta(q^6)*eta(q^10)*eta(q^15)) in powers of q. - Michael Somos, Sep 22 2005.
G.f.: product_{k>0}((1+x^k)*(1+x^(15k)))/((1+x^(3k))*(1+x^(5k))).
Euler transform of period 30 sequence [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, ...]. - Michael Somos, Sep 22 2005
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u*(u-v^2)^2 +v*(v-u^2)^2 -u*v -(u*v)^3. - Michael Somos, Sep 22 2005
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=(v+u*w)^2 -v*(u^2+w^2). - Michael Somos, Sep 22 2005
G.f.: Product_{k>0} (1+x^k-x^(3k)-x^(4k)-x^(5k)+x^(7k)+x^(8k)). - Michael Somos Sep 22 2005
a(n) ~ exp(2*Pi*sqrt(2*n/5)/3) / (2^(3/4) * sqrt(3) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
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EXAMPLE
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E.g. a(15)=5 because we can write 15 as 14+1=13+2=11+4=8+7=8+4+2+1.
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MAPLE
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series(product((1+x^k)*(1+x^(15*k))/((1+x^(3*k))*(1+x^(5*k))), k=1..100), x=0, 100);
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MATHEMATICA
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CoefficientList[ Series[ Product[(1 + x^k)(1 + x^(15*k))/((1 + x^(3k))*(1 + x^(5k))), {k, 100}], {x, 0, 75}], x] (* Robert G. Wilson v, Feb 22 2005 *)
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PROG
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(PARI) {a(n)=local(A); if (n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)*eta(x^5+A)*eta(x^30+A)/ (eta(x+A)*eta(x^6+A)*eta(x^10+A)*eta(x^15+A)), n))} /* Michael Somos, Sep 22 2005 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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