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%I #11 May 19 2018 20:07:53
%S 1,5,37,434,7751,193101,6200872,242667316,11144759839,585318934391,
%T 34511211188637,2253285826341378,161201686356627524,
%U 12530568505972885004,1051099249634285619168,94603882448795669308980,9092091650779263675187695,929177036869575506758681035,100608724821944458615599713935,11504982932704269804549116593702,1385525417578463389730054278506959
%N O.g.f. A(x) satisfies: [x^n] exp( n*(n+4) * x ) / A(x) = 0 for n>0.
%e O.g.f.: A(x) = 1 + 5*x + 37*x^2 + 434*x^3 + 7751*x^4 + 193101*x^5 + 6200872*x^6 + 242667316*x^7 + 11144759839*x^8 + 585318934391*x^9 + 34511211188637*x^10 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp(n*(n+4)*x) / A(x) begins:
%e n=0: [1, -5, -24, -1134, -100608, -14542200, -3095496000, -907608905280, ...];
%e n=1: [1, 0, -49, -1744, -128763, -17383400, -3572628125, -1024052930280, ...];
%e n=2: [1, 7, 0, -2430, -189600, -22895928, -4410982656, -1218708054720, ...];
%e n=3: [1, 16, 207, 0, -250107, -33107544, -5910144669, -1540910769048, ...];
%e n=4: [1, 27, 680, 13970, 0, -42775928, -8486494016, -2090851421760, ...];
%e n=5: [1, 40, 1551, 56376, 1681797, 0, -10852876125, -2994692165280, ...];
%e n=6: [1, 55, 2976, 156546, 7748832, 316211400, 0, -3807596825280, ...];
%e n=7: [1, 72, 5135, 360920, 24718725, 1597879072, 85448027299, 0, ...];
%e n=8: [1, 91, 8232, 738450, 65376768, 5650680456, 462123838656, 31350065660352, 0, ...]; ...
%e RELATED SERIES.
%e The logarithmic derivative of A(x) yields:
%e A'(x)/A(x) = 5 + 49*x + 872*x^2 + 22661*x^3 + 759915*x^4 + 30843448*x^5 + 1459277062*x^6 + 78529473925*x^7 + 4724556111179*x^8 + 313739794874469*x^9 + ...
%e 1 - 1/A(x) = 5*x + 12*x^2 + 189*x^3 + 4192*x^4 + 121185*x^5 + 4299300*x^6 + 180081132*x^7 + 8675950464*x^8 + 471853727865*x^9 + 28563862383700*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+3) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
%o for(n=0, 25, print1( a(n), ", "))
%Y Cf. A304322, A304318, A304319, A304863, A304864.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 19 2018
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