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

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

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

Example

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.

Prog

(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), ", "))

Crossrefs

Cf. A360582, A360583.

Sequence in context: A181197 A217807 A348651 * A127770 A121630 A089470

Adjacent sequences: A360581 A360582 A360583 * A360585 A360586 A360587

Keyword

nonn

Author

Paul D. Hanna, Mar 12 2023

Status

approved