A317341 E.g.f. A(x) satisfies: [x^n] 1/(1-x)^(n^2) / A(x)^n = 0 for n >= 1.

1, 1, 5, 73, 3025, 267761, 39973381, 8864616265, 2712723956993, 1090880803227745, 556508515320800101, 350642617901691170441, 267174141889391279470225, 242012057463863136292454353, 256946945274110281614185744165, 315958380772526173969828356406921, 445411627825937393526248850060842881, 713506179952629917462979541328787054785

Offset

0,3

Comments

It is remarkable that the logarithm of the e.g.f. A(x) is an integer series.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..216

Formula

a(n) ~ c * d^n * (n!)^2 / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.0711316736966... - Vaclav Kotesovec, Aug 10 2018

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 3025*x^4/4! + 267761*x^5/5! + 39973381*x^6/6! + 8864616265*x^7/7! + 2712723956993*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in 1/(1-x)^(n^2) / A(x)^n begins:
n=1: [1, 0, -3, -58, -2679, -249156, -38055995, -8542203342, ...];
n=2: [1, 2, 0, -128, -6328, -555552, -82280384, -18170728480, ...];
n=3: [1, 6, 33, 0, -11295, -1046358, -145984383, -31019236524, ...];
n=4: [1, 12, 144, 1520, 0, -1699104, -252074048, -50777317056, ...];
n=5: [1, 20, 405, 8050, 138665, 0, -387421475, -83789021650, ...];
n=6: [1, 30, 912, 27792, 824616, 21065184, 0, -124201808352, ...];
n=7: [1, 42, 1785, 76412, 3262497, 135099678, 4801008121, 0, ...];
n=8: [1, 56, 3168, 180640, 10339520, 588664512, 32441206912, 1531609302656, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 2*x^2 + 10*x^3 + 113*x^4 + 2091*x^5 + 53071*x^6 + 1699097*x^7 + 65414637*x^8 + 2935593649*x^9 + 150229832066*x^10 + ... + A317342(n)*x^n + ...

Prog

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

Crossrefs

Cf. A317342.

Sequence in context: A355122 A370542 A334282 * A012640 A353042 A128889

Adjacent sequences: A317338 A317339 A317340 * A317342 A317343 A317344

Keyword

nonn

Author

Paul D. Hanna, Jul 26 2018

Status

approved