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A338179
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G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} ( 1 + (1+x)^(n+k) ).
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3
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1, 2, 5, 18, 78, 389, 2171, 13293, 88234, 628645, 4772159, 38367795, 325108942, 2891485361, 26898627654, 260948353868, 2633065768017, 27571255842516, 298990713911228, 3351804623743392, 38780482844720442, 462405765265412815, 5674524701481226121, 71581950972419135866, 927174066925836999755
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + (1+x)^(n+k)).
G.f.: Sum_{n>=0} x^n * (1+x)^(n*(3*n-1)/2) / (Product_{k=0..n} 1 - x*(1+x)^(n+k)).
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 78*x^4 + 389*x^5 + 2171*x^6 + 13293*x^7 + 88234*x^8 + 628645*x^9 + 4772159*x^10 + 38367795*x^11 + 325108942*x^12 + ...
where
A(x) = 1 + x*(1 + (1+x)) + x^2*(1 + (1+x)^2)*(1 + (1+x)^3) + x^3*(1 + (1+x)^3)*(1 + (1+x)^4)*(1 + (1+x)^5) + x^4*(1 + (1+x)^4)*(1 + (1+x)^5)*(1 + (1+x)^6)*(1 + (1+x)^7) + x^5*(1 + (1+x)^5)*(1 + (1+x)^6)*(1 + (1+x)^7)*(1 + (1+x)^8)*(1 + (1+x)^9) + ... + x^n*Product_{k=0..n-1} (1 + (1+x)^(n+k)) + ...
Also
A(x) = 1/(1 - x) + x*(1+x)/((1 - x*(1+x))*(1 - x*(1+x)^2)) + x^2*(1+x)^5/((1 - x*(1+x)^2)*(1 - x*(1+x)^3)*(1 - x*(1+x)^4)) + x^3*(1+x)^12/((1 - x*(1+x)^3)*(1 - x*(1+x)^4)*(1 - x*(1+x)^5)*(1 - x*(1+x)^6)) + x^4*(1+x)^22/((1 - x*(1+x)^4)*(1 - x*(1+x)^5)*(1 - x*(1+x)^6)*(1 - x*(1+x)^7)*(1 - x*(1+x)^8)) + ... + x^n*(1+x)^(n*(3*n-1)/2)/(Product_{k=0..n} 1 - x*(1+x)^(n+k)) + ...
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MATHEMATICA
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nmax = 25; CoefficientList[Series[Sum[x^n * Product[1 + (1+x)^(n+k), {k, 0, n-1}], {n, 0, nmax-1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 15 2020 *)
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PROG
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(PARI) {a(n) = polcoeff( sum(m=0, n, x^m * prod(k=0, m-1, 1 + (1+x)^(m+k) +x*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = polcoeff( sum(m=0, n, x^m * (1+x +x*O(x^n))^(m*(3*m-1)/2) / prod(k=0, m, 1 - x*(1+x)^(m+k) +x*O(x^n)) ), 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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