A320669 O.g.f. A(x) satisfies: [x^n] exp(-n^3*A(x)) / (1 - n^2*x)^n = 0, for n > 0.

1, 2, 27, 2176, 316125, 92433420, 38689900249, 24036220587520, 19705732103751309, 21228545767337495500, 28631298365231328948940, 47701162183511368703635200, 95797470923250302955913961043, 228907109818475997814838969598324, 641132565508623116202107427900402750, 2082400670957118326405938988144017645568

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..16.

Example

O.g.f.: A(x) = x + 2*x^2 + 27*x^3 + 2176*x^4 + 316125*x^5 + 92433420*x^6 + 38689900249*x^7 + 24036220587520*x^8 + 19705732103751309*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^3*A(x)) / (1 - n^2*x)^n begins:
n=1: [1, 0, -3, -160, -52191, -37930176, -66549456875, ...];
n=2: [1, 0, 0, -1040, -414720, -303430848, -532404700160, ...];
n=3: [1, 0, 135, 0, -1237275, -1019993472, -1799293659165, ...];
n=4: [1, 0, 768, 22400, 0, -2155144704, -4259850874880, ...];
n=5: [1, 0, 2625, 136000, 25862625, 0, -7511859284375, ...];
n=6: [1, 0, 6912, 524880, 192513024, 36792874944, 0, ...];
n=7: [1, 0, 15435, 1591520, 938926485, 280095248832, 121196964253015, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m^3*x*Ser(A))/(1-m^2*x +x^2*O(x^m))^m)[m+1]/m^3 ); A[n]}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A320418, A320668, A319939.

Sequence in context: A320417 A113094 A327128 * A299116 A076113 A182335

Adjacent sequences: A320666 A320667 A320668 * A320670 A320671 A320672

Keyword

nonn

Author

Paul D. Hanna, Oct 19 2018

Status

approved