A317342 - OEIS

%I #12 Oct 14 2018 08:03:25

%S 1,2,10,113,2091,53071,1699097,65414637,2935593649,150229832066,

%T 8625636578142,548722452568519,38292095950421691,2907573526178822138,

%U 238606864933942697326,21041901922349773004745,1984311337163146655202793,199256107170401946445319128,21226198257061414084824564640,2390892224585501751308482295221,283921776093206079595272233555143

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

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

%H Vaclav Kotesovec, <a href="/A317342/b317342.txt">Table of n, a(n) for n = 1..340</a>

%F a(n) ~ c * d^n * n! / 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

%e O.g.f.: 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 + ...

%e such that [x^n] 1/(1-x)^(n^2) / exp( n*A(x) )  = 0  for n >= 1.

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in 1/(1-x)^(n^2) / exp( n*A(x) ) begins:

%e n=1: [1, 0, -3, -58, -2679, -249156, -38055995, -8542203342, ...];

%e n=2: [1, 2, 0, -128, -6328, -555552, -82280384, -18170728480, ...];

%e n=3: [1, 6, 33, 0, -11295, -1046358, -145984383, -31019236524, ...];

%e n=4: [1, 12, 144, 1520, 0, -1699104, -252074048, -50777317056, ...];

%e n=5: [1, 20, 405, 8050, 138665, 0, -387421475, -83789021650, ...];

%e n=6: [1, 30, 912, 27792, 824616, 21065184, 0, -124201808352, ...];

%e n=7: [1, 42, 1785, 76412, 3262497, 135099678, 4801008121, 0, ...];

%e n=8: [1, 56, 3168, 180640, 10339520, 588664512, 32441206912, 1531609302656, 0, ...]; ...

%e in which the coefficient of x^n in row n forms a diagonal of zeros.

%e RELATED SERIES.

%e exp(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! + ... + A317341(n)*x^n/n! + ...

%o (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 ); polcoeff( log(Ser(A)),n)}

%o for(n=1,20,print1(a(n),", "))

%Y Cf. A317341, A319833, A319835, A319836, A319832.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 26 2018