A319939 O.g.f. A(x) satisfies: [x^n] exp(-n^2*A(x)) / (1 - n*x)^n = 0, for n > 0.

1, 1, 3, 24, 325, 6642, 176204, 5828160, 228372291, 10374419250, 534203188948, 30762752950224, 1956914341159778, 136286437739608492, 10310240639621093400, 841935232438747348480, 73807352585103519962815, 6913603998931859925828282, 689148541231545351838902508, 72838943589708142133363904400, 8137053663063956034586144506558, 958035702236154579666369909892724

Offset

1,3

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

Formula

a(n) ~ c * d^n * n! / n^3, where d = 6.1103392... and c = 0.05165... - Vaclav Kotesovec, Oct 24 2020

Example

O.g.f.: A(x) = x + x^2 + 3*x^3 + 24*x^4 + 325*x^5 + 6642*x^6 + 176204*x^7 + 5828160*x^8 + 228372291*x^9 + 10374419250*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^2*A(x)) / (1 - n*x)^n begins:
n=1: [1, 0, -1, -16, -567, -38816, -4771025, -886931424, ...];
n=2: [1, 0, 0, -40, -2112, -154464, -19097600, -3549131520, ...];
n=3: [1, 0, 9, 0, -3483, -333504, -43269795, -8050921776, ...];
n=4: [1, 0, 32, 224, 0, -454016, -75031040, -14515172352, ...];
n=5: [1, 0, 75, 800, 21225, 0, -92559125, -22271154000, ...];
n=6: [1, 0, 144, 1944, 88128, 2515104, 0, -25624491264, ...];
n=7: [1, 0, 245, 3920, 252693, 10516576, 505622425, 0, ...];
n=8: [1, 0, 384, 7040, 602112, 30829056, 2210682880, 134210187264, 0, ...];
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2 + 25*x^3/3! + 673*x^4/4! + 42501*x^5/5! + 5048251*x^6/6! + 924544573*x^7/7! + 242568147585*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^2*x*Ser(A))/(1-m*x +x^2*O(x^m))^m)[m+1]/m^2 ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A319938, A319940, A320669.

Sequence in context: A264561 A003236 A232693 * A082166 A354259 A370055

Adjacent sequences: A319936 A319937 A319938 * A319940 A319941 A319942

Keyword

nonn

Author

Paul D. Hanna, Oct 09 2018

Status

approved