A360582 Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^n) = 0 for n > 0.
1, 1, 2, 8, 48, 382, 3793, 45208, 627957, 9928646, 175476102, 3420270423, 72789704826, 1678446235555, 41675807453127, 1108522434288617, 31444611938560078, 947522959703143140, 30225484159719768548, 1017558928058932606182, 36053690169955373601165, 1341103168079733579768368
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 48*x^4 + 382*x^5 + 3793*x^6 + 45208*x^7 + 627957*x^8 + 9928646*x^9 + 175476102*x^10 + ... The table of coefficients in the successive powers of g.f. A(x) begins: n = 1: [1, 1, 2, 8, 48, 382, 3793, 45208, ...]; n = 2: [1, 2, 5, 20, 116, 892, 8606, 100298, ...]; n = 3: [1, 3, 9, 37, 210, 1566, 14687, 167280, ...]; n = 4: [1, 4, 14, 60, 337, 2448, 22340, 248580, ...]; n = 5: [1, 5, 20, 90, 505, 3591, 31935, 347120, ...]; n = 6: [1, 6, 27, 128, 723, 5058, 43919, 466410, ...]; n = 7: [1, 7, 35, 175, 1001, 6923, 58828, 610653, ...]; ... The table of coefficients in A(x)/(1 + x*A(x)^n) begins: n = 1: [1, 0, 1, 5, 34, 293, 3066, 37900, ...]; n = 2: [1, 0, 0, 3, 25, 235, 2601, 33346, ...]; n = 3: [1, 0, -1, 0, 14, 167, 2055, 28049, ...]; n = 4: [1, 0, -2, -4, 0, 89, 1432, 21994, ...]; n = 5: [1, 0, -3, -9, -18, 0, 742, 15216, ...]; n = 6: [1, 0, -4, -15, -41, -102, 0, 7820, ...]; n = 7: [1, 0, -5, -22, -70, -220, -775, 0, ...]; ... in which the diagonal of all zeros illustrates that [x^n] A(x) / (1 + x*A(x)^n) = 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-1)), #A-1) );A[n+1]} for(n=0,30,print1(a(n),", "))
Formula
a(n) ~ c * n! * n^(2*LambertW(1) - 1) / LambertW(1)^n, where c = 0.11249164340900724981958... - Vaclav Kotesovec, Mar 13 2023