A204452 A014330 - A203577. Difference between the exponential convolution of A000108 (Catalan) with itself and the corresponding exponential half-convolution.
0, 1, 2, 11, 34, 212, 804, 5567, 24014, 178148, 839596, 6501420, 32658872, 259775440, 1368965576, 11080668871, 60613092662, 496461841956, 2798385807012, 23113333523180, 133539494791000, 1109722749130576, 6545965568001272
Offset: 0
Examples
With A000108 = {1, 1, 2, 5, 14, 42,...}
a(4) = 4*5*1 + 1*14*1 = 34.
a(5) = 10*5*2 + 5*14*1 + 1*42*1 = 212.
Programs
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Maple
A204452 := proc(n) add( binomial(n,k)*A000108(k)*A000108(n-k), k=floor(n/2)+1..n) ; end proc: seq(A204452(n),n=0..50) ; # R. J. Mathar, Jul 27 2022
Formula
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.
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.
C(x) = hypergeom([1/2],[2],4*x) (see the e.g.f. of A000108 for the version involving BesselI functions), and
C2(x) = hypergeom([1/2,1/2],[1,2,2],16*x).
Comments