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

0,3

Links

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

Formula

a(n) ~ c * n! * n^(2*LambertW(1) - 1) / LambertW(1)^n, where c = 0.11249164340900724981958... - Vaclav Kotesovec, Mar 13 2023

Example

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.

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. A360581, A360583, A360584.

Sequence in context: A219613 A124453 A211827 * A322339 A000165 A241122

Adjacent sequences: A360579 A360580 A360581 * A360583 A360584 A360585

Keyword

nonn

Author

Paul D. Hanna, Mar 12 2023

Status

approved