A360583 Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^(n+1)) = 0 for n > 0.
1, 1, 3, 17, 139, 1455, 18326, 267700, 4426686, 81455357, 1646941293, 36238989035, 861298646217, 21978627323651, 599195351716464, 17379759869328515, 534392854606942358, 17363705283005593096, 594513962872698955686, 21395116871674867310280, 807416275879430472577570
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 139*x^4 + 1455*x^5 + 18326*x^6 + 267700*x^7 + 4426686*x^8 + 81455357*x^9 + 1646941293*x^10 + ... The table of coefficients in the successive powers of g.f. A(x) begins: n = 1: [1, 1, 3, 17, 139, 1455, 18326, ...]; n = 2: [1, 2, 7, 40, 321, 3290, 40685, ...]; n = 3: [1, 3, 12, 70, 555, 5583, 67827, ...]; n = 4: [1, 4, 18, 108, 851, 8424, 100624, ...]; n = 5: [1, 5, 25, 155, 1220, 11916, 140085, ...]; n = 6: [1, 6, 33, 212, 1674, 16176, 187372, ...]; n = 7: [1, 7, 42, 280, 2226, 21336, 243817, ...]; ... The table of coefficients in A(x)/(1 + x*A(x)^(n+1)) begins: n = 1: [1, 0, 1, 9, 88, 1021, 13736, 209940, ...]; n = 2: [1, 0, 0, 5, 64, 821, 11670, 184622, ...]; n = 3: [1, 0, -1, 0, 35, 587, 9283, 155666, ...]; n = 4: [1, 0, -2, -6, 0, 315, 6555, 122855, ...]; n = 5: [1, 0, -3, -13, -42, 0, 3467, 86025, ...]; n = 6: [1, 0, -4, -21, -92, -364, 0, 45079, ...]; n = 7: [1, 0, -5, -30, -151, -784, -3866, 0, ...]; ... in which the diagonal of all zeros illustrates that [x^n] A(x) / (1 + x*A(x)^(n+1)) = 0 for n > 0.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = -polcoeff( Ser(A)/(1 + x*Ser(A)^(#A)), #A-1) );A[n+1]} for(n=0,30,print1(a(n),", "))
Formula
a(n) ~ c * n! * n^(3*LambertW(1) - 1 + 1/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.0685246538334926708088... - Vaclav Kotesovec, Mar 13 2023