A321085 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, 36, 330, 4092, 63308, 1165952, 24802704, 596862420, 16003332092, 472588688880, 15231153085304, 531801227022912, 19992702649780800, 805099660439308672, 34574148236596003008, 1577229602738525889984, 76172611622917782581040, 3882814240467405224924960, 208333046141920439476582608, 11737307286279509352537747760

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 - n*x^2) = 0, for n > 0, when G(x) = x + x^2 + x*G(x)*G'(x), where G(x)/x is the o.g.f. of A321086.

Links

Table of n, a(n) for n=1..22.

Example

O.g.f.: A(x) = x + 2*x^2 + 6*x^3 + 36*x^4 + 330*x^5 + 4092*x^6 + 63308*x^7 + 1165952*x^8 + 24802704*x^9 + 596862420*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, -819, -39056, -2923925, -317422764, ...];
n=2: [1, 0, 0, -32, -1392, -75552, -5832320, -635767680, ...];
n=3: [1, 0, 3, 0, -1323, -100008, -8542665, -955410984, ...];
n=4: [1, 0, 8, 80, 0, -89024, -10215680, -1248268032, ...];
n=5: [1, 0, 15, 220, 3405, 0, -8752325, -1409888100, ...];
n=6: [1, 0, 24, 432, 9936, 234144, 0, -1176833664, ...];
n=7: [1, 0, 35, 728, 20853, 710248, 23232055, 0, ...];
n=8: [1, 0, 48, 1120, 37632, 1560192, 72348160, 3135469056, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 1081*x^4/4! + 46001*x^5/5! + 3272701*x^6/6! + 345526945*x^7/7! + 50126588849*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 25*x^3/3! - 695*x^4/4! - 34401*x^5/5! - 2665019*x^6/6! - 295314937*x^7/7! - 44140455855*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. A319938, A321086.

Sequence in context: A061302 A055541 A275551 * A376517 A133822 A133892

Adjacent sequences: A321082 A321083 A321084 * A321086 A321087 A321088

Keyword

nonn

Author

Paul D. Hanna, Oct 27 2018

Status

approved