A317341 - OEIS

%I #8 Aug 10 2018 06:31:13

%S 1,1,5,73,3025,267761,39973381,8864616265,2712723956993,

%T 1090880803227745,556508515320800101,350642617901691170441,

%U 267174141889391279470225,242012057463863136292454353,256946945274110281614185744165,315958380772526173969828356406921,445411627825937393526248850060842881,713506179952629917462979541328787054785

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

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

%H Vaclav Kotesovec, <a href="/A317341/b317341.txt">Table of n, a(n) for n = 0..216</a>

%F 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

%e 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! + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in 1/(1-x)^(n^2) / A(x)^n 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 LOGARITHMIC PROPERTY.

%e The logarithm of the e.g.f. is the integer series:

%e 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 + ...

%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 ); n!*A[n+1]}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A317342.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 26 2018