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A238882
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Coefficients in a variant of Ramanujan's wrong identity for prime number partitions.
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1
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1, 1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -2, 2, -1, 3, 1, 2, 1, -1, -4, 1, -4, -4, -10, -2, -8, -4, -5, -4, -1, 1, 2, 5, 6, 13, 12, 16, 18, 21, 25, 23, 30, 22, 23, 21, 21, 18, 14, 8, -1, -9, -20, -36, -36, -51, -61, -75, -80, -96, -103
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OFFSET
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0,16
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COMMENTS
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Consider the g.f. of the prime parts partition numbers, GF = 1/Product_{k>=1} (1-x^prime(k)), cf. A000607. Then consecutively subtract a(n)*x^b(n)/Product_{k=1..n} (1-x^k), n=0,1,2,3,... where a(n)*x^b(n) is the leading term of the remaining expression, GF - previously subtracted terms. Sequence A238804 lists the exponents b(n), here we list the coefficients a(n).
The identity Ramanujan considered, GF = Sum_{n>=0} x^Sum_{k=1..n} prime(k)/Product_{k=1..n} (1-x^k), or A000607 = A046676, is wrong: In the way they are defined above, the pattern of b(n) = (sum of first n primes) breaks after b(4)=17; the pattern a(n)=1 breaks also after n=4 (which yields this sequence), and the nontrivial cancellations stop after the power b(5)=21, followed by 22, 24, 25, 26, 27, ...
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LINKS
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EXAMPLE
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GF = 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)...) = 1+x^2+x^3+x^4+2*x^5+... (cf. A000607)
=> a(0)=1, b(0)=0, GF - 1 = x^2 + ....
=> a(1)=1, b(1)=2, GF - 1 - x^2/(1-x) = x^5 + ...
=> a(2)=1, b(2)=5, GF - 1 - x^2/(1-x) - x^5/(1-x)(1-x^2) = x^10 + ...
=> a(3)=1, b(3)=10, GF - ... - x^10/(1-x)(1-x^2)(1-x^3) = x^17 + ...
=> a(4)=1, b(4)=17, GF - ... - x^17/(1-x)(1-x^2)(1-x^3)(1-x^4) = -x^21+...
=> a(5)=-1, b(5)=21, GF - ... + x^21/... etc.
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PROG
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(PARI) p=1/prod(k=1, 25, 1-x^prime(k), 1+O(x^999))/* Note: p1+...+p25 > 1000 */; for(k=0, 99, print1(polcoeff(p, c=valuation(p, x)), ", "); p-=polcoeff(p, c)*x^c/prod(j=1, k, 1-x^j, O(x^199)+1))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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