|
%I #15 Aug 01 2018 23:10:27
%S 1,1,0,-3,-5,10,58,23,-557,-1421,4094,28316,-52,-449150,-970286,
%T 5908939,31046627,-49583353,-750617284,-544416915,15819383275,
%U 46795708732,-288245326872,-1808819140124,3784215933076,57664747490276,14416027504376,-1664155475303224,-3937904190952656,43893853942734810,219165998056699650
%N O.g.f. A(x) satisfies: [x^n] exp( n*x*A(x) ) * (n+1 - n*A(x)) = 0 for n >= 1.
%C Compare: the factorial series F(x) = Sum_{n>=0} n!*x^n satisfies
%C (1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n >= 1,
%C (2) [x^n] exp( n*x*F(x) ) * (2 - F(x)) = 0 for n >= 1,
%C (3) [x^n] exp( n^2*x*F(x) ) * (n + 1 - n*F(x)) = 0 for n >= 1.
%C It is remarkable that this sequence should consist entirely of integers.
%H Paul D. Hanna, <a href="/A317338/b317338.txt">Table of n, a(n) for n = 0..400</a>
%e O.g.f.: A(x) = 1 + x - 3*x^3 - 5*x^4 + 10*x^5 + 58*x^6 + 23*x^7 - 557*x^8 - 1421*x^9 + 4094*x^10 + 28316*x^11 - 52*x^12 - 449150*x^13 - 970286*x^14 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp( n*x*A(x) ) * (n+1 - n*A(x)) begins:
%e n=1: [1, 0, 1, 16, 117, -704, -35075, -200304, 17660041, ...];
%e n=2: [1, 0, 0, 20, 288, 912, -51200, -888480, 19165440, ...];
%e n=3: [1, 0, -3, 0, 333, 3888, -27135, -1471824, 4665465, ...];
%e n=4: [1, 0, -8, -56, 0, 5344, 33280, -1317312, -15647744, ...];
%e n=5: [1, 0, -15, -160, -1035, 0, 81325, -180000, -25008375, ...];
%e n=6: [1, 0, -24, -324, -3168, -20304, 0, 1156896, -10209024, ...];
%e n=7: [1, 0, -35, -560, -6867, -67088, -422975, 0, 19205305, ...];
%e n=8: [1, 0, -48, -880, -12672, -155712, -1525760, -9408384, 0, ...];
%e n=9: [1, 0, -63, -1296, -21195, -305856, -3806595, -37346832, -230393079, 0, ...]; ...
%e in which the coefficients of x^n in row n form a diagonal of zeros.
%o (PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*Ser(A) ) * (m - (m-1)*Ser(A)) )[m]/(m-1) ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A317337, A305110, A305114, A305115, A305116.
%K sign
%O 0,4
%A _Paul D. Hanna_, Aug 01 2018
|
