A320415 O.g.f. A(x) satisfies: [x^n] 1/A(x)^(n+1) / (1 - (n+1)*x) = 0, for n > 0.

1, 1, 2, 7, 41, 365, 4429, 67631, 1235436, 26136471, 626438616, 16743921605, 493193891312, 15860617916787, 552723118104318, 20743853592796035, 834059197889873217, 35767193981175943493, 1629529714740523133093, 78603418440584895849957, 4002217080821288857968629, 214514119185341611087477513, 12073658983251570372687654623

Offset

0,3

Links

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

Formula

a(n) ~ c * n^(n-1), where c = 0.9132042556231... - Vaclav Kotesovec, Aug 11 2021

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 41*x^4 + 365*x^5 + 4429*x^6 + 67631*x^7 + 1235436*x^8 + 26136471*x^9 + 626438616*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in 1/A(x)^n / (1 - n*x) begins
n=1: [1, 0, -1, -5, -33, -314, -3972, -62229, ...];
n=2: [1, 0, -1, -8, -63, -624, -7930, -124272, ...];
n=3: [1, 0, 0, -7, -81, -909, -11908, -187260, ...];
n=4: [1, 0, 2, 0, -69, -1072, -15530, -250656, ...];
n=5: [1, 0, 5, 15, 0, -896, -17415, -307260, ...];
n=6: [1, 0, 9, 40, 162, 0, -14317, -333048, ...];
n=7: [1, 0, 14, 77, 462, 2205, 0, -269991, ...];
n=8: [1, 0, 20, 128, 954, 6560, 36152, 0, ...];
n=9: [1, 0, 27, 195, 1701, 14202, 110790, 691857, 0, ...]; ...
in which the coefficient of x^(n-1) in row n forms a diagonal of zeros, after an initial '1'.
RELATED SERIES.
1/A(x) = 1 - x - x^2 - 4*x^3 - 28*x^4 - 281*x^5 - 3658*x^6 - 58257*x^7 - 1093228*x^8 - 23558599*x^9 - 572266776*x^10 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( 1/Ser(A)^m / (1 - m*x +x^2*O(x^m)) )[m]/m ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A319945.

Sequence in context: A346271 A101390 A113144 * A006846 A047864 A173916

Adjacent sequences: A320412 A320413 A320414 * A320416 A320417 A320418

Keyword

nonn

Author

Paul D. Hanna, Oct 12 2018

Status

approved