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A255930 Expansion of exp( Sum_{n >= 1} A210674(n)*x^n/n ).

%I A255930 #7 Mar 01 2025 01:18:02
%S A255930 1,3,33,991,63060,7018860,1206748720,295775068680,97835325011235,
%T A255930 41970842737399345,22655642596496388759,15025240474194493147857,
%U A255930 12008582230377080862401692,11382727559611560650861409564,12625404970864692720119281536900,16199644066580777034289339157904220
%N A255930 Expansion of exp( Sum_{n >= 1} A210674(n)*x^n/n ).
%C A255930 It appears that this sequence is integer valued.
%C A255930 The o.g.f. A(x) = 1 + 3*x + 33*x^2 + 991*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A210674.
%C A255930 This sequence is the particular case m = 3 of the following general conjecture.
%C A255930 Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
%C A255930 For cases see A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928 (m = 1) and A255929(m = 2).
%C A255930 Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.
%F A255930 O.g.f.: exp(3*x + 57*x^2/2 + 2703*x^3/3 + 239277*x^4/4 + ...) = 1 + 3*x + 33*x^2 + 991*x^3 + 63060*x^4 + ....
%F A255930 a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} A210674(n-k)*a(k) for n >= 1.
%p A255930 #A255930
%p A255930 A210674 := proc (n) option remember; if n = 0 then 1 else 3*add(binomial(2*n, 2*k)*A210674(k), k = 0 .. n-1) end if; end proc:
%p A255930 A255930 := proc (n) option remember; if n = 0 then 1 else add(A210674(n-k)*A255930(k), k = 0 .. n-1)/n end if; end proc:
%p A255930 seq(A255930(n), n = 0 .. 15);
%Y A255930 Cf. A210674, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1), A255929(m = 2).
%K A255930 nonn,easy
%O A255930 0,2
%A A255930 _Peter Bala_, Mar 11 2015