A320414 O.g.f. A(x) satisfies: [x^n] exp(-n*A(x)) / (1 - n^2*x) = 0 for n >= 1.

1, 5, 144, 10082, 1264535, 245875182, 68150171838, 25482852420656, 12358294214448753, 7544949396113515915, 5664150492647564303056, 5129806105907894893467492, 5516524251630079831171874313, 6950350571025359814277640201432, 10142321626803167978417939290871040, 16972169807104759800475307306764090752, 32287069027792648627362032680785777091413

Offset

1,2

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Example

G.f.: A(x) = x + 5*x^2 + 144*x^3 + 10082*x^4 + 1264535*x^5 + 245875182*x^6 + 68150171838*x^7 + 25482852420656*x^8 + 12358294214448753*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) / (1 - n^2*x) begins:
n=1: [1, 0, -9, -862, -241719, -151666596, -176990046545, ...];
n=2: [1, 2, 0, -1616, -495232, -308287632, -357675223424, ...];
n=3: [1, 6, 87, 0, -693639, -475046118, -548265525633, ...];
n=4: [1, 12, 360, 14240, 0, -586861344, -749452608128, ...];
n=5: [1, 20, 975, 69430, 5820185, 0, -861802226825, ...];
n=6: [1, 30, 2136, 226368, 31268736, 4762503504, 0, ...];
n=7: [1, 42, 4095, 597044, 115492713, 27293766318, 6830535431671, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 11*x^2/2! + 895*x^3/3! + 245785*x^4/4! + 153050681*x^5/5! + 177988541251*x^6/6! + 344758772825671*x^7/7! + 1030280671456569905*x^8/8! + ...
exp(-A(x)) = 1 - x - 9*x^2/2! - 835*x^3/3! - 238271*x^4/4! - 150458001*x^5/5! - 176080046969*x^6/6! - 342201963425491*x^7/7! - 1024667102754203775*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^2*x +x^2*O(x^m)))[m+1]/m ); A[n]}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A300617.

Sequence in context: A222289 A134503 A282836 * A168041 A081322 A232986

Adjacent sequences: A320411 A320412 A320413 * A320415 A320416 A320417

Keyword

nonn

Author

Paul D. Hanna, Oct 26 2018

Status

approved