A204452 - OEIS

%I #7 Jul 27 2022 08:34:06

%S 0,1,2,11,34,212,804,5567,24014,178148,839596,6501420,32658872,

%T 259775440,1368965576,11080668871,60613092662,496461841956,

%U 2798385807012,23113333523180,133539494791000,1109722749130576,6545965568001272

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

%C For the exponential (also known as binomial) half-convolution of the Catalan sequence A000108 with itself see A203577.

%F 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 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     C(x) = hypergeom([1/2],[2],4*x) (see the e.g.f. of A000108 for the version involving BesselI functions), and

%F     C2(x) = hypergeom([1/2,1/2],[1,2,2],16*x).

%e With A000108 = {1, 1, 2, 5, 14, 42,...}

%e   a(4) = 4*5*1 + 1*14*1 = 34.

%e   a(5) = 10*5*2 + 5*14*1 + 1*42*1 = 212.

%p A204452 := proc(n)

%p     add( binomial(n,k)*A000108(k)*A000108(n-k), k=floor(n/2)+1..n) ;

%p end proc:

%p seq(A204452(n),n=0..50) ; # _R. J. Mathar_, Jul 27 2022

%Y Cf. A000108, A014330, A203577.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Jan 16 2012