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A338184
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * A(x)^(n+1) /( Product_{k=1..n+1} 1 + x^k*A(x) ).
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2
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1, 2, 6, 20, 72, 274, 1088, 4462, 18768, 80552, 351452, 1554360, 6953182, 31406472, 143043384, 656225618, 3029606032, 14065166648, 65623654844, 307542991080, 1447064884324, 6833469156024, 32375938380454, 153853312669856, 733137984581920, 3502378475368244, 16770811951363082, 80479149475376288
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OFFSET
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0,2
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COMMENTS
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Compare the g.f. to the following series identities.
(1) 1 = Sum_{n>=0} x^(n*(n+1)/2) * a^n /( Product_{k=1..n+1} 1 + x^k*a ), which holds for all a.
(2) Q = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) /( Product_{k=1..n+1} 1 + x^k ),
(3) Q = Sum_{n>=0} (-1)^n * x^n * ( Product_{k=0..n-1} 1 + x^k ),
where Q = 1 - 2*Sum_{n>=1} (x^(n*(3*n-1)/2) - x^(n*(3*n+1)/2)) = 1 - 2*x + 2*x^2 - 2*x^5 + 2*x^7 - 2*x^12 + 2*x^15 - 2*x^22 + 2*x^26 + ...
For n>0, a(n) = 2 (mod 4) iff n is a generalized pentagonal number (A001318), and a(n) = 0 (mod 4) elsewhere (conjecture).
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * A(x)^(n+1) /( Product_{k=1..n+1} 1 + x^k*A(x) ).
(2) 1 = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^n /( Product_{k=1..n+1} 1 + x^k*A(x) ).
(3) 1 = Sum_{n>=0} (-1)^n * x^n * A(x)^(n+1) * ( Product_{k=0..n-1} 1 + x^k ).
a(n) ~ c * d^n / n^(3/2), where d = 5.074478094065946942699247453461914358319648... and c = 1.036722248866192132778767799931330371... - Vaclav Kotesovec, Oct 17 2020
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 72*x^4 + 274*x^5 + 1088*x^6 + 4462*x^7 + 18768*x^8 + 80552*x^9 + 351452*x^10 + ...
where, if we set A = A(x), then
1 = A/(1 + x*A) - x*A^2/((1 + x*A)*(1 + x^2*A)) + x^3*A^3/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)) - x^6*A^4/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)) + x^10*A^5/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)*(1 + x^5*A)) -+ ...
also (trivially),
1 = 1/(1 + x*A) + x*A/((1 + x*A)*(1 + x^2*A)) + x^3*A^2/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)) + x^6*A^3/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)) + x^10*A^4/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)*(1 + x^5*A)) + ...
Further,
1 = A - 2*x*A^2 + 2*x^2*A^3*(1+x) - 2*x^3*A^4*(1+x)*(1+x^2) + 2*x^4*A^5*(1+x)*(1+x^2)*(1+x^3)*(1+x^4) - 2*x^5*A^6*(1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5) + 2*x^6*A^7*(1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)*(1+x^6) -+ ... + (-1)^n * x^n * A(x)^(n+1) * Product_{k=0..n-1} (1 + x^k) + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = -Vec( sum(m=0, sqrtint(2*n+1), (-1)^m * x^(m*(m+1)/2) * Ser(A)^(m+1) / prod(k=1, m+1, 1 + x^k*Ser(A))) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^(n+1) * Product_{k=0..n-1} (1 + x^k) */
{a(n) = my(A=1); for(i=1, n, A = 1 - 2*sum(m=1, n, (-x)^m * A^(m+1) * prod(k=1, m-1, 1 + x^k +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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