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A204452 A014330 - A203577. Difference between the exponential convolution of A000108 (Catalan) with itself and the corresponding exponential half-convolution.

%I A204452 #7 Jul 27 2022 08:34:06
%S A204452 0,1,2,11,34,212,804,5567,24014,178148,839596,6501420,32658872,
%T A204452 259775440,1368965576,11080668871,60613092662,496461841956,
%U A204452 2798385807012,23113333523180,133539494791000,1109722749130576,6545965568001272
%N A204452 A014330 - A203577. Difference between the exponential convolution of A000108 (Catalan) with itself and the corresponding exponential half-convolution.
%C A204452 For the exponential (also known as binomial) half-convolution of the Catalan sequence A000108 with itself see A203577.
%F A204452 a(n) = sum(binomial(n,k)*C(k)*C(n-k),k=floor(n/2)+1..n), n>=0, with C(n)=A000108(n), the Catalan numbers.
%F A204452 E.g.f.: (C(x)^2 - C2(x^2))/2 with the e.g.f. C(x) of A000108, and the o.g.f. C2(x) of the sequence {(C(n)/n!)^2}. Compare this with the e.g.f. of A203577.
%F A204452     C(x) = hypergeom([1/2],[2],4*x) (see the e.g.f. of A000108 for the version involving BesselI functions), and
%F A204452     C2(x) = hypergeom([1/2,1/2],[1,2,2],16*x).
%e A204452 With A000108 = {1, 1, 2, 5, 14, 42,...}
%e A204452   a(4) = 4*5*1 + 1*14*1 = 34.
%e A204452   a(5) = 10*5*2 + 5*14*1 + 1*42*1 = 212.
%p A204452 A204452 := proc(n)
%p A204452     add( binomial(n,k)*A000108(k)*A000108(n-k), k=floor(n/2)+1..n) ;
%p A204452 end proc:
%p A204452 seq(A204452(n),n=0..50) ; # _R. J. Mathar_, Jul 27 2022
%Y A204452 Cf. A000108, A014330, A203577.
%K A204452 nonn,easy
%O A204452 0,3
%A A204452 _Wolfdieter Lang_, Jan 16 2012