A110156
G.f.: A(x) = Product_{n>=1} 1/(1 - 4^n*x^n)^(2/4^n); self-convolution equals A110154.
Original entry on oeis.org
1, 2, 8, 26, 106, 350, 1512, 5110, 21476, 77886, 319148, 1141038, 4910266, 17499058, 72541048, 272237050, 1121013506, 4112829790, 17377874692, 63697436318, 265450712278, 1003409368250, 4102752994248, 15321419162722, 64725434306768
Offset: 0
A(x) = 1 + 2*x + 8*x^2 + 26*x^3 + 106*x^4 + 350*x^5 +... =
1/[(1-4*x)^(2/4)*(1-16*x^2)^(2/16)*(1-64*x^3)^(2/64)*...].
A110152
G.f.: A(x) = Product_{n>=1} 1/(1 - 2^n*x^n)^(2/2^n).
Original entry on oeis.org
1, 2, 6, 14, 36, 78, 192, 406, 942, 2018, 4512, 9450, 21178, 43950, 95532, 200398, 431356, 892518, 1917572, 3950614, 8410230, 17398466, 36648980, 75326754, 159199004, 326471706, 683028924, 1404145162, 2930071798, 5993625942
Offset: 0
G.f.: A(x) = 1 + 2*x + 6*x^2 + 14*x^3 + 36*x^4 + 78*x^5 +...
where
A(x) = 1/((1-2*x) * (1-4*x^2)^(1/2) * (1-8*x^3)^(1/4) * (1-16*x^4)^(1/8) *...).
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nmax = 30; CoefficientList[Series[Product[1/(1 - 2^k*x^k)^(2/2^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
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a(n)=polcoeff(prod(k=1,n,1/(1-2^k*x^k+x*O(x^n))^(2/2^k)),n)
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A090879(n) = sumdiv(n,d, d*2^(n-d))
a(n)=local(A);A=exp(sum(k=1,n,2*A090879(k)*x^k/k)+x*O(x^n));polcoeff(A,n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Jan 05 2014
A110153
Expansion of g.f.: Product_{n>=1} 1/(1 - 3^n*x^n)^(3/3^n).
Original entry on oeis.org
1, 3, 12, 39, 138, 426, 1461, 4458, 14655, 45309, 145479, 443037, 1427196, 4329696, 13655325, 41795679, 131102229, 397649811, 1247247507, 3775785681, 11761535064, 35770717695, 110693177805, 335003030301, 1040296817955, 3145674794979, 9695067728493, 29405519846121
Offset: 0
A(x) = 1 + 3*x + 12*x^2 + 39*x^3 + 138*x^4 + 426*x^5 + ... =
1/[(1-3*x)*(1-9*x^2)^(1/3)*(1-27*x^3)^(1/9)*(1-81*x^4)^(1/27)*...].
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nmax=27; CoefficientList[Series[Product[1/(1 - 3^n*x^n)^(3/3^n),{n,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Jun 21 2024 *)
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a(n)=polcoeff(prod(k=1,n,1/(1-3^k*x^k+x*O(x^n))^(3/3^k)),n)
A110155
G.f.: A(x) = Product_{n>=1} 1/(1 - 5^n*x^n)^(5/5^n).
Original entry on oeis.org
1, 5, 30, 155, 855, 4305, 23255, 116705, 614655, 3108355, 16168430, 81136755, 422279580, 2116775030, 10893334980, 54857686305, 281413208380, 1410806289330, 7237582729155, 36262232611605, 185184341144805
Offset: 0
A(x) = 1 + 5*x + 30*x^2 + 155*x^3 + 855*x^4 + 4305*x^5 +... =
1/[(1-5*x)*(1-25*x^2)^(1/5)*(1-125*x^3)^(1/25)*(1-625*x^4)^(1/125)*...]
Showing 1-4 of 4 results.