A319832 O.g.f. A(x) satisfies: [x^n] (1+x)^(n^3) / exp( n^2*A(x) ) = 0 for n >= 1.

1, 1, 64, 7431, 1715876, 655110323, 372565312967, 294961986584223, 310044461434740982, 417747271801946191399, 702143923935041675022156, 1440542259736400467662865991, 3543917937671552230039341072619, 10300091517989144989707891748204281, 34923380149186824044791021305086297101, 136645283241569758193338091605962393536127

Offset

1,3

Comments

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

Links

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

Example

G.f.: A(x) = x + x^2 + 64*x^3 + 7431*x^4 + 1715876*x^5 + 655110323*x^6 + 372565312967*x^7 + 294961986584223*x^8 + ...
The table of coefficients of x^k/k! in (1+x)^(n^3)/exp(n^2*A(x)) begins
n=1: [1, 0, -3, -382, -178323, -205893636, -471669948095, ...];
n=2: [1, 4, 0, -1648, -738256, -837882624, -1906457557376, ...];
n=3: [1, 18, 279, 0, -1826631, -2007298314, -4452097032657, ...];
n=4: [1, 48, 2208, 90752, 0, -3956975616, -8595954537728, ...];
n=5: [1, 100, 9825, 938150, 81392525, 0, -14962864950875, ...];
n=6: [1, 180, 32112, 5663088, 977957712, 154892058624, 0, ...];
n=7: [1, 294, 85995, 25005092, 7212994089, 2046788086266, 535857247292899, 0, ...]; ...
in which the coefficient of x^n in the n-th row forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 391*x^3/3! + 179905*x^4/4! + 206808441*x^5/5! + 472924417411*x^6/6! + 1881046356701023*x^7/7! + ...

Prog

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

Crossrefs

Cf. A319831, A319833.

Sequence in context: A180377 A013995 A264007 * A063410 A075417 A294083

Adjacent sequences: A319829 A319830 A319831 * A319833 A319834 A319835

Keyword

nonn

Author

Paul D. Hanna, Sep 28 2018

Status

approved