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A338181
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G.f.: A(x) satisfies 1 = Sum_{n>=0} Product_{k=2*n..3*n-1} ( (1+x)^k - A(x) ).
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3
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1, 2, 7, 123, 3434, 127389, 5800285, 310671959, 19065484061, 1316721595692, 101007119697478, 8520008641904667, 783858935294161761, 78131288902786790863, 8389163022856205911508, 965540248030539267674460, 118603148642277195505854462, 15489151634920585316431414382, 2143263988200596035777277393467
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OFFSET
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0,2
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COMMENTS
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Compare the g.f. A(x) to functions B(x) and C(x) defined by
A338178: 1 = Sum_{n>=0} Product_{k=n..2*n-1} ((1+x)^k - B(x)) ;
A338183: 1 = Sum_{n>=0} Product_{k=3*n..4*n-1} ((1+x)^k - C(x)).
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} Product_{k=2*n..3*n-1} ( (1+x)^k - A(x) ).
(2) 1 = Sum_{n>=0} (1+x)^(n*(5*n-1)/2) / Product_{k=2*n..3*n} (1 + (1+x)^k*A(x)).
a(n) ~ c * d^n * n^n / exp(n), where d = 7.8565736454589... and c = 0.30167199686... - Vaclav Kotesovec, Aug 12 2021
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 7*x^2 + 123*x^3 + 3434*x^4 + 127389*x^5 + 5800285*x^6 + 310671959*x^7 + 19065484061*x^8 + 1316721595692*x^9 + 101007119697478*x^10 + ...
Let A = A(x), then g.f. A(x) satisfies
1 = 1 + ((1+x)^2 - A) + ((1+x)^4 - A)*((1+x)^5 - A) + ((1+x)^6 - A)*((1+x)^7 - A)*((1+x)^8 - A) + ((1+x)^8 - A)*((1+x)^9 - A)*((1+x)^10 - A)*((1+x)^11 - A) + ((1+x)^10 - A)*((1+x)^11 - A)*((1+x)^12 - A)*((1+x)^13 - A)*((1+x)^14 - A) + ... + Product_{k=2*n..3*n-1} ((1+x)^k - A(x)) + ...
also
1 = 1/(1 + A) + (1+x)^2/((1 + (1+x)^2*A)*(1 + (1+x)^3*A)) + (1+x)^9/((1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)) + (1+x)^21/((1 + (1+x)^6*A)*(1 + (1+x)^7*A)*(1 + (1+x)^8*A)*(1 + (1+x)^9*A)) + (1+x)^38/((1 + (1+x)^8*A)*(1 + (1+x)^9*A)*(1 + (1+x)^10*A)*(1 + (1+x)^11*A)*(1 + (1+x)^12*A)) + ... + (1+x)^(n*(5*n-1)/2)/( Product_{k=2*n..3*n} (1 + (1+x)^k*A(x)) ) + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, prod(k=2*m, 3*m-1, (1+x)^k - Ser(A)) ) )[#A] ); A[n+1]}
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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