A360584 Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^(n+2)) = 0 for n > 0.
1, 1, 4, 29, 294, 3727, 55748, 950898, 18094313, 378363501, 8600306451, 210773059751, 5534376088000, 154911828439188, 4603267204022882, 144710918709587399, 4798300212740184379, 167370947204751098624, 6127130537038980726113, 234905895680130694945861, 9413383171884998924237972
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 294*x^4 + 3727*x^5 + 55748*x^6 + 950898*x^7 + 18094313*x^8 + 378363501*x^9 + 8600306451*x^10 + ... The table of coefficients in the successive powers of g.f. A(x) begins: n = 1: [1, 1, 4, 29, 294, 3727, 55748, ...]; n = 2: [1, 2, 9, 66, 662, 8274, 122143, ...]; n = 3: [1, 3, 15, 112, 1116, 13776, 200827, ...]; n = 4: [1, 4, 22, 168, 1669, 20384, 293654, ...]; n = 5: [1, 5, 30, 235, 2335, 28266, 402710, ...]; n = 6: [1, 6, 39, 314, 3129, 37608, 530334, ...]; n = 7: [1, 7, 49, 406, 4067, 48615, 679140, ...]; ... The table of coefficients in A(x)/(1 + x*A(x)^(n+2)) begins: n = 1: [1, 0, 1, 13, 166, 2391, 38776, 699060, ...]; n = 2: [1, 0, 0, 7, 119, 1911, 32823, 612983, ...]; n = 3: [1, 0, -1, 0, 64, 1358, 26039, 515774, ...]; n = 4: [1, 0, -2, -8, 0, 724, 18356, 406634, ...]; n = 5: [1, 0, -3, -17, -74, 0, 9702, 284785, ...]; n = 6: [1, 0, -4, -27, -159, -824, 0, 149478, ...]; n = 7: [1, 0, -5, -38, -256, -1759, -10833, 0, ...]; ... in which the diagonal of all zeros illustrates that [x^n] A(x) / (1 + x*A(x)^(n+2)) = 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^(4*LambertW(1) - 1 + 2/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.02048373460253911846... - Vaclav Kotesovec, Mar 13 2023