%I A131980
%S A131980 1,2,6,2,24,24,120,240,24,720,2400,720,5040,25200,15120,720
%N A131980 A coefficient tree from the list partition transform relating A000165, 
               A110327, A000129, A110330 and A000142.
%C A131980 Construct the infinite array of polynomials
%C A131980 a(0,t) = 1
%C A131980 a(1,t) = 2
%C A131980 a(2,t) = 6 + 2 t
%C A131980 a(3,t) = 24 + 24 t
%C A131980 a(4,t) = 120 + 240 t + 24 t^2
%C A131980 a(5,t) = 720 + 2400 t + 720 t^2
%C A131980 a(6,t) =5040 + 25200 t + 15120 t^2 + 720 t^3
%C A131980 This array is the reciprocal array of the following array b(n,t) under 
               the list partition transform and its associated operations described 
               in A133314.
%C A131980 b(0,t) = 1, b(1,t) = -2, b(2,t) = -2*(t-1), b(n,t) = 0 for n>2 .
%C A131980 Then A000165(n) = a(n,1) .
%C A131980 Lower triangular matrix A110327 = binomial(n,k)*a(n-k,2) .
%C A131980 n! * A000129(n+1) = a(n,2) = A110327(n,0) .
%C A131980 A110330 = matrix inverse of binomial(n,k)*a(n-k,2) = binomial(n,k)*b(n-k,
               2) .
%C A131980 A000142(n+1) = a(n,0) .
%F A131980 exp[b(.,t)*x] = 1 - 2x - (t-1) * x^2; therefore exp[a(.,t)*x] = 1 / [ 
               1 - 2x - (t-1) * x^2 ] = (t-1) / { t - [ 1 + x*(t-1) ]^2 }.
%F A131980 Also, a(n,t) = (1 - t*u^2)^(n+1) (D_u)^n [ 1 / (1 - t*u^2) ] with eval. 
               at u = 1/t . Compare A076743.
%F A131980 a(n,t) = n! sum(k=0,1,...) binomial(n+1,2k+1) * t^k = n! sum(k=0,1,...) 
               A034867(n,k) * t^k .
%F A131980 E.g.f. = I_o[2*(u*x)^(1/2)] * exp[a(.,t)*x], analogous to A132382,
%F A131980 where I_o is the zeroth modified Bessel function of the first kind, i.e. 
               I_o[2*(u*x)^(1/2)] = sum(j=0,1,...) u^j/j! * x^j/j! .
%F A131980 Additional relations are given by formulae in A133314.
%F A131980 a(n,t) = n! sum(k=0,...,n) binomial(n+1,2k+1) * t^k = n! sum(k=0,...,
               n) A034867(n,k) * t^k .
%Y A131980 Sequence in context: A008556 A096485 A125032 this_sequence A076743 A141056 
               A027760
%Y A131980 Adjacent sequences: A131977 A131978 A131979 this_sequence A131981 A131982 
               A131983
%K A131980 easy,nonn
%O A131980 0,2
%A A131980 Tom Copeland (tcjpn(AT)msn.com), Oct 30 2007, Nov 29 2007, Nov 30 2007