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A338183
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G.f.: A(x) satisfies 1 = Sum_{n>=0} Product_{k=3*n..4*n-1} ( (1+x)^k - A(x) ).
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3
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1, 3, 15, 355, 13876, 722211, 46171804, 3473185910, 299362847750, 29037782693087, 3128420024771954, 370593699771993304, 47880982409336565169, 6701959229159656143832, 1010495137429274173541486, 163310611718381473156298904, 28168286488482397457218340375, 5165418587961263257650366514583
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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)) ;
A338181: 1 = Sum_{n>=0} Product_{k=2*n..3*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=3*n..4*n-1} ( (1+x)^k - A(x) ).
(2) 1 = Sum_{n>=0} (1+x)^(n*(7*n-1)/2) / Product_{k=3*n..4*n} (1 + (1+x)^k*A(x)).
a(n) ~ c * d^n * n^n / exp(n), where d = 11.0309918920494... and c = 0.314387643322... - Vaclav Kotesovec, Aug 12 2021
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 15*x^2 + 355*x^3 + 13876*x^4 + 722211*x^5 + 46171804*x^6 + 3473185910*x^7 + 299362847750*x^8 + 29037782693087*x^9 + ...
Let A = A(x), then g.f. A(x) satisfies
1 = 1 + ((1+x)^3 - A) + ((1+x)^6 - A)*((1+x)^7 - A) + ((1+x)^9 - A)*((1+x)^10 - A)*((1+x)^11 - A) + ((1+x)^12 - A)*((1+x)^13 - A)*((1+x)^14 - A)*((1+x)^15 - A) + ((1+x)^15 - A)*((1+x)^16 - A)*((1+x)^17 - A)*((1+x)^18 - A)*((1+x)^19 - A) + ... + Product_{k=3*n..4*n-1} ((1+x)^(n+k) - A(x)) + ...
also
1 = 1/(1 + A) + (1+x)^3/((1 + (1+x)^3*A)*(1 + (1+x)^4*A)) + (1+x)^13/((1 + (1+x)^6*A)*(1 + (1+x)^7*A)*(1 + (1+x)^8*A)) + (1+x)^30/((1 + (1+x)^9*A)*(1 + (1+x)^10*A)*(1 + (1+x)^11*A)*(1 + (1+x)^12*A)) + (1+x)^54/((1 + (1+x)^12*A)*(1 + (1+x)^13*A)*(1 + (1+x)^14*A)*(1 + (1+x)^15*A)*(1 + (1+x)^16*A)) + ... + (1+x)^(n*(7*n-1)/2)/( Product_{k=3*n..4*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=3*m, 4*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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