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A338180
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G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} ( 1 + (1+x)^(2*n+k) ).
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2
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1, 2, 6, 27, 152, 982, 7170, 57745, 506426, 4787074, 48377495, 519377787, 5892707464, 70345730334, 880325589031, 11511926195652, 156873359666502, 2222244083243157, 32654408362860528, 496783112039964012, 7811234596327634934, 126743609828281136160, 2119210893034560256558
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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)^(2*n+k)).
G.f.: Sum_{n>=0} x^n * (1+x)^(n*(5*n-1)/2) / ( Product_{k=0..n} 1 - x*(1+x)^(2*n+k) ).
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 6*x^2 + 27*x^3 + 152*x^4 + 982*x^5 + 7170*x^6 + 57745*x^7 + 506426*x^8 + 4787074*x^9 + 48377495*x^10 + ...
where
A(x) = 1 + x*(1 + (1+x)^2) + x^2*(1 + (1+x)^4)*(1 + (1+x)^5) + x^3*(1 + (1+x)^6)*(1 + (1+x)^7)*(1 + (1+x)^8) + x^4*(1 + (1+x)^8)*(1 + (1+x)^9)*(1 + (1+x)^10)*(1 + (1+x)^11) + x^5*(1 + (1+x)^10)*(1 + (1+x)^11)*(1 + (1+x)^12)*(1 + (1+x)^13)*(1 + (1+x)^14) + ... + x^n*Product_{k=0..n-1} (1 + (1+x)^(2*n+k)) + ...
Also
A(x) = 1/(1 - x) + x*(1+x)^2/((1 - x*(1+x)^2)*(1 - x*(1+x)^3)) + x^2*(1+x)^9/((1 - x*(1+x)^4)*(1 - x*(1+x)^5)*(1 - x*(1+x)^6)) + x^3*(1+x)^21/((1 - x*(1+x)^6)*(1 - x*(1+x)^7)*(1 - x*(1+x)^8)*(1 - x*(1+x)^9)) + x^4*(1+x)^38/((1 - x*(1+x)^8)*(1 - x*(1+x)^9)*(1 - x*(1+x)^10)*(1 - x*(1+x)^11)*(1 - x*(1+x)^12)) + ... + x^n*(1+x)^(n*(5*n-1)/2)/(Product_{k=0..n} 1 - x*(1+x)^(2*n+k)) + ...
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MATHEMATICA
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nmax = 25; CoefficientList[Series[Sum[x^n * Product[1 + (1+x)^(2*n+k), {k, 0, n-1}], {n, 0, nmax-1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 16 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)^(2*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*(5*m-1)/2) / prod(k=0, m, 1 - x*(1+x)^(2*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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