A321086 O.g.f. A(x) satisfies: [x^n] exp(n*A(x)) * (1 - n*x - n*x^2) = 0, for n > 0.

1, 2, 6, 32, 220, 1812, 17108, 180512, 2093760, 26396160, 358741328, 5223336288, 81079811280, 1336407320080, 23311138957200, 429063111959808, 8311760620707648, 169072470759431232, 3603666131945918144, 80327823251439861760, 1869212211081119135616, 45331401566332423284864, 1143967734536203174726784, 29996686272924492809481216, 816185909551276017516640000

Offset

1,2

Comments

It is remarkable that this sequence should consist entirely of integers.

Compare to: [x^n] exp(n*G(x)) * (1 - n*x) = 0, for n > 0, when G(x) = x + x*G(x)*G'(x), where G(x)/x is the o.g.f. of A088716.

Links

Table of n, a(n) for n=1..25.

Formula

O.g.f. A(x) satisfies: A(x) = x + x^2 + x*A(x)*A'(x).

Example

O.g.f.: A(x) = x + 2*x^2 + 6*x^3 + 32*x^4 + 220*x^5 + 1812*x^6 + 17108*x^7 + 180512*x^8 + 2093760*x^9 + 26396160*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) * (1 - n*x - n*x^2) begins:
n=1: [1, 0, 1, 28, 729, 26416, 1321225, 87466716, ...];
n=2: [1, 0, 0, 32, 1200, 49152, 2569600, 172974720, ...];
n=3: [1, 0, -3, 0, 1089, 60408, 3509325, 246760776, ...];
n=4: [1, 0, -8, -80, 0, 49024, 3777280, 293683968, ...];
n=5: [1, 0, -15, -220, -2535, 0, 2848825, 291386100, ...];
n=6: [1, 0, -24, -432, -7056, -105984, 0, 208089216, ...];
n=7: [1, 0, -35, -728, -14175, -293048, -5733875, 0, ...];
n=8: [1, 0, -48, -1120, -24576, -590592, -15603200, -391709184, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
(a) Differential Equation.
O.g.f. A(x) satisfies: A(x) = x + x^2 + x*A(x)*A'(x) where
A'(x) = 1 + 4*x + 18*x^2 + 128*x^3 + 1100*x^4 + 10872*x^5 + 119756*x^6 + ...
A(x)*A'(x) = x + 6*x^2 + 32*x^3 + 220*x^4 + 1812*x^5 + 17108*x^6 + 17108*x^7 + ...
so that A(x) - x*A(x)*A'(x) = x + x^2.
(b) Exponentiation.
exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 985*x^4/4! + 32321*x^5/5! + 1544701*x^6/6! + 99637105*x^7/7! + 8257877489*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 25*x^3/3! - 599*x^4/4! - 21681*x^5/5! - 1106939*x^6/6! - 74873737*x^7/7! - 6431021295*x^8/8! + ...

Prog

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

Crossrefs

Cf. A088716, A321085, A321087.

Sequence in context: A224884 A361354 A357664 * A111550 A261398 A009686

Adjacent sequences: A321083 A321084 A321085 * A321087 A321088 A321089

Keyword

nonn

Author

Paul D. Hanna, Oct 27 2018

Status

approved