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%I #8 May 19 2018 20:07:27
%S 1,4,26,288,5012,122608,3869456,148838816,6721823600,347434618432,
%T 20180665251360,1299399587904384,91769604540962816,
%U 7049102617933604352,584848346900868001792,52109481481410100183552,4961586770799906448318208,502707358017324652042259456,54000226687663791374322245120,6129804668943947684749062516736,733179029209444818691965317379072
%N O.g.f. A(x) satisfies: [x^n] exp( n*(n+3) * x ) / A(x) = 0 for n>0.
%e O.g.f.: A(x) = 1 + 4*x + 26*x^2 + 288*x^3 + 5012*x^4 + 122608*x^5 + 3869456*x^6 + 148838816*x^7 + 6721823600*x^8 + 347434618432*x^9 + 20180665251360*x^10 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp(n*(n+3)*x) / A(x) begins:
%e n=0: [1, -4, -20, -864, -72576, -10060800, -2068882560, -589017945600, ...];
%e n=1: [1, 0, -36, -1232, -89088, -11667456, -2328963200, -650497926144, ...];
%e n=2: [1, 6, 0, -1664, -125136, -14853600, -2803074560, -757869964800, ...];
%e n=3: [1, 14, 160, 0, -162000, -20768352, -3651775488, -937273259520, ...];
%e n=4: [1, 24, 540, 10000, 0, -26468352, -5107476608, -1241737082880, ...];
%e n=5: [1, 36, 1260, 41536, 1133184, 0, -6460818560, -1740188582400, ...];
%e n=6: [1, 50, 2464, 118368, 5374512, 202760544, 0, -2192436486144, ...];
%e n=7: [1, 66, 4320, 279136, 17619504, 1054101600, 52553405440, 0, ...];
%e n=8: [1, 84, 7020, 582400, 47760000, 3832731648, 292170316672, 18603667330560, 0, ...]; ...
%e RELATED SERIES.
%e The logarithmic derivative of A(x) yields:
%e A'(x)/A(x) = 4 + 36*x + 616*x^2 + 15496*x^3 + 504624*x^4 + 19947072*x^5 + 921521248*x^6 + 48536700064*x^7 + 2864002270720*x^8 + 186878075521216*x^9 + ...
%e 1 - 1/A(x) = 4*x + 10*x^2 + 144*x^3 + 3024*x^4 + 83840*x^5 + 2873448*x^6 + 116868640*x^7 + 5488631808*x^8 + 291890096640*x^9 + 17321970359200*x^10 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m+2) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
%o for(n=0, 25, print1( a(n), ", "))
%Y Cf. A304322, A304318, A304319, A304863, A304865.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 19 2018
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