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A293603
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G.f.: Product_{n=-oo..+oo} (1 + x^n*(1 + x^n)^n).
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3
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4, 2, 12, 6, 20, 14, 62, 46, 108, 74, 244, 176, 486, 364, 956, 680, 1682, 1326, 3300, 2522, 5884, 4562, 10328, 8312, 18680, 15028, 31472, 25832, 54888, 45676, 92828, 78142, 156158, 133688, 258464, 225832, 429172, 376450, 699878, 624618, 1140294, 1031538, 1845002, 1681284, 2949698, 2729894, 4725338, 4409132, 7485858, 7048744, 11811642, 11269324, 18553170
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OFFSET
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0,1
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LINKS
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FORMULA
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G.f.: A(x) = P(x) * Q(x) where
P(x) = Product_{n>=0} ( 1 + x^n*(1 + x^n)^n ),
Q(x) = Product_{n>=1} ( 1 + x^(n^2-n)/(1 + x^n)^n ).
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EXAMPLE
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G.f.: A(x) = 4 + 2*x + 12*x^2 + 6*x^3 + 20*x^4 + 14*x^5 + 62*x^6 + 46*x^7 + 108*x^8 + 74*x^9 + 244*x^10 + 176*x^11 + 486*x^12 + 364*x^13 + 956*x^14 + 680*x^15 + 1682*x^16 + 1326*x^17 + 3300*x^18 + 2522*x^19 + 5884*x^20 + 4562*x^21 + 10328*x^22 + 8312*x^23 + 18680*x^24 + 15028*x^25 + 31472*x^26 + 25832*x^27 + 54888*x^28 + 45676*x^29 + 92828*x^30 +...
where
A(x) = P(x) * Q(x) where
P(x) = 2 * (1 + x*(1+x)) * (1 + x^2*(1+x^2)^2) * (1 + x^3*(1+x^3)^3) * (1 + x^4*(1+x^4)^4) * (1 + x^5*(1+x^5)^5) *...* ( 1 + x^n*(1 + x^n)^n ) *...
Q(x) = (1 + 1/(1+x)) * (1 + x^2/(1+x^2)^2) * (1 + x^6/(1+x^3)^3) * (1 + x^12/(1+x^4)^4) * (1 + x^20/(1+x^5)^5) *...* (1 + x^(n^2-n)/(1 + x^n)^n ) *...
Explicitly,
P(x) = 2 + 2*x + 4*x^2 + 4*x^3 + 10*x^4 + 12*x^5 + 22*x^6 + 26*x^7 + 46*x^8 + 56*x^9 + 98*x^10 + 118*x^11 + 192*x^12 + 232*x^13 + 368*x^14 + 458*x^15 + 686*x^16 + 848*x^17 + 1294*x^18 + 1604*x^19 + 2312*x^20 + 2888*x^21 + 4160*x^22 + 5200*x^23 + 7322*x^24 + 9120*x^25 + 12694*x^26 + 15890*x^27 + 21982*x^28 + 27392*x^29 + 37304*x^30 +...
Q(x) = 2 - x + 3*x^2 - 2*x^3 - 2*x^4 + 8*x^6 - 4*x^7 - 2*x^8 - 7*x^9 + 10*x^10 - 13*x^11 + 18*x^12 - 2*x^13 + 28*x^14 - 56*x^15 - 5*x^16 - 12*x^17 + 86*x^18 - 40*x^19 + 53*x^20 - 121*x^21 + 87*x^22 - 96*x^23 + 105*x^24 - 40*x^25 + 137*x^26 - 259*x^27 + 147*x^28 - 238*x^29 + 401*x^30 +...
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MATHEMATICA
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terms = 53; Product[If[n == 0, 2, (1 + x^n*(1 + x^n)^n)], {n, -terms, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Nov 04 2017 *)
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PROG
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(PARI) {a(n) = my(A); A = prod(m=-n-1, n+1, (1 + x^m*(1 + x^m)^m + x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 60, 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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