A183606 G.f. satisfies: [x^(n+1)] A(x)^n = n*(n+1)*{[x^n] A(x)^n} for n>=0.

1, 1, 2, 13, 176, 3886, 125374, 5550713, 323184220, 23969935266, 2208185433708, 247518073493022, 33184572994243884, 5244964339235985636, 965302981114301621022, 204679102209593395617929

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Formula

G.f.: A(x) = x/Series_Reversion(G(x)) where A(x*G(x)) = G(x) satisfies: G(x) = 1/(1-x - x*{d/dx x^2*G'(x)/G(x)}) and is the g.f. of A183607.

a(n) ~ c * n!^2 / n, where c = 2.05242598709856847249724356196772... - Vaclav Kotesovec, Oct 22 2020

Example

G.f.: A(x) = 1 + x + 2*x^2 + 13*x^3 + 176*x^4 + 3886*x^5 +...
A(x) satisfies A(x*G(x)) = G(x) where:
G(x) = 1 + x + 3*x^2 + 20*x^3 + 249*x^4 + 5087*x^5 + 155180*x^6 +...
which satisfies: G(x) = 1/(1-x - x*{d/dx x^2*G'(x)/G(x)}).
...
The table of coefficients in the initial powers of g.f. A(x) begins:
A^0: [1,_0, 0, 0, 0, 0, 0, 0, ...];
A^1: [1, 1,_2, 13, 176, 3886, 125374, 5550713, 323184220, ...];
A^2: [1, 2, 5,_30, 382, 8176, 259393, 11372294, 658103374, ...];
A^3: [1, 3, 9, 52,_624, 12921, 402749, 17479176, 1005197070, ...];
A^4: [1, 4, 14, 80, 909,_18180, 556210, 23886840, 1364926338, ...];
A^5: [1, 5, 20, 115, 1245, 24021,_720630, 30611920, 1737775110, ...];
A^6: [1, 6, 27, 158, 1641, 30522, 896960,_37672320, 2124251838, ...];
A^7: [1, 7, 35, 210, 2107, 37772, 1086260, 45087344,_2524891264, ...];
...
where the upper diagonal:
[0, 2, 30, 624, 18180, 720630, 37672320, 2524891264, ...]
is term-wise related to the main diagonal in the following way:
[0*1, 2*1, 6*5, 12*52, 20*909, 30*24021, 42*896960, 56*45087344, ...].

Prog

(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=((#A-1)*(#A-2)*Vec(Ser(A)^(#A-2))[#A-1] - Vec(Ser(A)^(#A-2))[#A])/(#A-2)); A[n+1]}

Crossrefs

Cf. A183607.

Sequence in context: A119400 A182314 A268988 * A366194 A307655 A137610

Adjacent sequences: A183603 A183604 A183605 * A183607 A183608 A183609

Keyword

nonn

Author

Paul D. Hanna, Jan 13 2011

Status

approved