A337579 - OEIS

%I #5 Sep 02 2020 04:49:33

%S 1,1,5,202,25741,6481768,2661785172,1606979708104,1336018641201031,

%T 1461946920710738032,2036450966030220362632,3519187269661662800713808,

%U 7390652429852470066011519746,18545709306030399397877283499248,54823008100459892066683079355901888,188621839026471088419358039473633535392

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

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

%e O.g.f.: A(x) = 1 + x + 5*x^2 + 202*x^3 + 25741*x^4 + 6481768*x^5 + 2661785172*x^6 + 1606979708104*x^7 + 1336018641201031*x^8 + ...

%e ILLUSTRATION OF DEFINITION.

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

%e n=0: [1, -1, -8, -1158, -607824, -771471360, -1906996245120, ...];

%e n=1: [1, 0, -36, -4736, -2447112, -3096809856, -7645376634080, ...];

%e n=2: [1, 9, 0, -12114, -5911488, -7219467792, -17580593299968, ...];

%e n=3: [1, 32, 880, 0, -12002784, -14133084672, -33100636472064, ...];

%e n=4: [1, 75, 5400, 341650, 0, -25227867600, -57875848640000, ...];

%e n=5: [1, 144, 20412, 2803392, 343375416, 0, -95154559008480, ...];

%e n=6: [1, 245, 59584, 14323974, 3357877488, 709290480864, 0, ...];

%e n=7: [1, 384, 146880, 55883776, 21079051392, 7789007628288, 2612787154865152, 0, ...]; ...

%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp(n*(n+1)^2*x) / A(x)^((n+1)^2) = 0 for n>0.

%e RELATED SERIES.

%e log(A(x)) = x + 9*x^2/2 + 592*x^3/3 + 102125*x^4/4 + 32276196*x^5/5 + 15931091190*x^6/6 + 11230009495552*x^7/7 + 10675195543084221*x^8/8 + ...

%e where [x^n] exp( (n+1)^2 * (n*x - log(A(x))) ) = 0 for n>0.

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

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

%Y Cf. A337458.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 01 2020