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%I #14 Mar 14 2018 06:19:39
%S 1,3,48,1510,71280,4511808,361640832,35516910960,4184770003200,
%T 582762638275840,94800017774905344,17836975939663156224,
%U 3847898790157443653632,944223655310222217584640,261663903298936561335828480,81353978185283974468642093056,28208743160867030634605718994944,10849126423364041648181194666082304,4605289001051501407092469612444385280
%N O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * A(x) ) = n*(n+1) * [x^(n-1)] exp( n*(n+1) * A(x) ) for n>=1.
%C Compare to: [x^n] exp( n*(n+1) * x ) = (n+1) * [x^(n-1)] exp( n*(n+1) * x ) for n>=1.
%C O.g.f. equals the logarithm of the e.g.f. of A300870.
%C The e.g.f. G(x) of A300870 satisfies: [x^n] G(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] G(x)^(n*(n+1)) for n>=1.
%C It is conjectured that this sequence consists entirely of integers.
%C a(n) is divisible by n*(n+1)/2 (conjecture); a(n) = n*(n+1)/2 * A300872(n).
%H Paul D. Hanna, <a href="/A300871/b300871.txt">Table of n, a(n) for n = 1..200</a>
%e O.g.f.: A(x) = x + 3*x^2 + 48*x^3 + 1510*x^4 + 71280*x^5 + 4511808*x^6 + 361640832*x^7 + 35516910960*x^8 + 4184770003200*x^9 + ...
%e where
%e exp(A(x)) = 1 + x + 7*x^2/2! + 307*x^3/3! + 37537*x^4/4! + 8755561*x^5/5! + 3304572391*x^6/6! + 1847063377867*x^7/7! + 1447456397632897*x^8/8! + ... + A300870(n)*x^n/n! + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in exp( n*(n+1) * A(x) ) begins:
%e n=1: [(1), (2), 8, 328/3, 9728/3, 2241184/15, 420248704/45, ...];
%e n=2: [1, (6), (36), 432, 11328, 2470464/5, 150254784/5, ...];
%e n=3: [1, 12, (108), (1296), 29136, 5776128/5, 335166336/5, ...];
%e n=4: [1, 20, 260, (10480/3), (209600/3), 7265600/3, 1173400640/9, ...];
%e n=5: [1, 30, 540, 8640, (166800), (5004000), 241367040, 116509893120/7...];
%e n=6: [1, 42, 1008, 19656, 396816, (53339328/5), (2240251776/5), ...];
%e n=7: [1, 56, 1736, 124096/3, 2767184/3, 355355392/15, (38932329856/45), (2180210471936/45), ...]; ...
%e in which the coefficients in parenthesis are related by
%e 2 = 1*2*(1); 36 = 2*3*(6); 1296 = 3*4*(108); 209600/3 = 4*5*(10480/3); 5004000 = 5*6*(166800); 2240251776/5 = 6*7*(53339328/5); ...
%e illustrating: [x^n] exp( n*(n+1) * A(x) ) = n*(n+1) * [x^(n-1)] exp( n*(n+1) * A(x) ).
%e The values A300872(n) = a(n) / (n*(n+1)/2) begin:
%e [1, 1, 8, 151, 4752, 214848, 12915744, 986580860, 92994888960, ...]
%e and appear to consist entirely of integers.
%o (PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)*(#A))); A[#A] = ((#A-1)*(#A)*V[#A-1] - V[#A])/(#A-1)/(#A) ); polcoeff( log(Ser(A)), n)}
%o for(n=1, 20, print1(a(n), ", "))
%Y Cf. A300870, A300872, A300591, A296171, A300874.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Mar 14 2018
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