A320415 - OEIS

%I #6 Aug 11 2021 16:35:36

%S 1,1,2,7,41,365,4429,67631,1235436,26136471,626438616,16743921605,

%T 493193891312,15860617916787,552723118104318,20743853592796035,

%U 834059197889873217,35767193981175943493,1629529714740523133093,78603418440584895849957,4002217080821288857968629,214514119185341611087477513,12073658983251570372687654623

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

%H Paul D. Hanna, <a href="/A320415/b320415.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ c * n^(n-1), where c = 0.9132042556231... - _Vaclav Kotesovec_, Aug 11 2021

%e 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 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in 1/A(x)^n / (1 - n*x) begins

%e n=1: [1, 0, -1, -5, -33, -314, -3972, -62229, ...];

%e n=2: [1, 0, -1, -8, -63, -624, -7930, -124272, ...];

%e n=3: [1, 0, 0, -7, -81, -909, -11908, -187260, ...];

%e n=4: [1, 0, 2, 0, -69, -1072, -15530, -250656, ...];

%e n=5: [1, 0, 5, 15, 0, -896, -17415, -307260, ...];

%e n=6: [1, 0, 9, 40, 162, 0, -14317, -333048, ...];

%e n=7: [1, 0, 14, 77, 462, 2205, 0, -269991, ...];

%e n=8: [1, 0, 20, 128, 954, 6560, 36152, 0, ...];

%e n=9: [1, 0, 27, 195, 1701, 14202, 110790, 691857, 0, ...]; ...

%e in which the coefficient of x^(n-1) in row n forms a diagonal of zeros, after an initial '1'.

%e RELATED SERIES.

%e 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 + ...

%o (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]}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A319945.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 12 2018