A319943 O.g.f. A(x) satisfies: [x^n] exp( n^4*x - n^3*A(x) ) = 0 for n >= 1.

1, 4, 756, 666368, 1573194000, 7732870168896, 68447889079649744, 991668063561584680960, 21964178018215273705692768, 706356702284500884771233120000, 31676677475982382039405057763384448, 1917371897548937365539316712575130025984, 152498179997818599198940803864398810967251456

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

Formula

a(n) ~ sqrt(1-c) * 2^(8*n - 7) * n^(3*n - 7/2) / (sqrt(Pi) * c^(n - 1/4) * (4-c)^(3*n - 3) * exp(3*n)), where c = -LambertW(-4*exp(-4)) = 0.079309605127113656439108647386463779474372628792... - Vaclav Kotesovec, Oct 13 2020

Example

G.f.: A(x) = x + 4*x^2 + 756*x^3 + 666368*x^4 + 1573194000*x^5 + 7732870168896*x^6 + 68447889079649744*x^7 + 991668063561584680960*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^4*x - n^3*A(x) ) begins:
n=1: [1, 0, -8, -4536, -15992640, -188782917120, ...];
n=2: [1, 8, 0, -37312, -129112064, -1515383749632, ...];
n=3: [1, 54, 2700, 0, -453396528, -5216886215136, ...];
n=4: [1, 192, 36352, 6492672, 0, -12944825376768, ...];
n=5: [1, 500, 249000, 122933000, 57869896000, 0, ...];
n=6: [1, 1080, 1164672, 1253133504, 1340717598720, 1376766221921280, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 9*x^2/2! + 4561*x^3/3! + 16011217*x^4/4! + 188863653441*x^5/5! + 5568801588349081*x^6/6! + 345016372861620194449*x^7/7! + ...

Prog

(PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^4*x +x*O(x^#A)) / Ser(A)^(m^3) )[m+1]/m^3 ); polcoeff( log(Ser(A)), n)}
for(n=1, 15, print1(a(n), ", "))

Crossrefs

Cf. A319941, A319942, A319944, A317344, A317346, A317347.

Sequence in context: A195625 A268838 A292306 * A181200 A308141 A284813

Adjacent sequences: A319940 A319941 A319942 * A319944 A319945 A319946

Keyword

nonn

Author

Paul D. Hanna, Oct 02 2018

Status

approved