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A203577
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Exponential (or binomial) half-convolution of the sequence A000108 (Catalan) with itself.
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2
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1, 1, 4, 11, 58, 212, 1304, 5567, 37734, 178148, 1284124, 6501420, 48758648, 259775440, 2000594288, 11080668871, 86930955662, 496461841956, 3947716126292, 23113333523180, 185660199980696, 1109722749130576, 8983793097101144, 54645629076275356, 445109373450545608, 2748480598104423952
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OFFSET
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0,3
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COMMENTS
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For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see A203576, where also the rule for the e.g.f. is given.
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LINKS
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Table of n, a(n) for n=0..25.
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FORMULA
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a(n) =
sum(binomial(n,k)*Catalan(k)*Catalan(n-k), k=0..floor(n/2)), n>=0.
E.g.f.: (C(x)^2 + C2(x^2))/2 with the e.g.f. C(x) of A000108, and the e.g.f. C2(x):=sum(Catalan(n)^2/n!*x^n/n!,n=0..infty) of the scaled Catalan squares. See a comment above.
C(x) = hypergeom([1/2],[2],4*x) (see A000108 for the version involving BesselI functions), and
C2(x) = hypergeom([1/2,1/2],[1,2,2],16*x).
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EXAMPLE
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With Catalan = A000108 = {1, 1, 2, 5, 14, 42,... }
a(4)= 1*1*14 + 4*1*5 + 6*2*2 = 58.
a(5)= 1*1*42 + 5*1*14 + 10*2*5 = 212..
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PROG
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(PARI) hat(b, n) = sum(k=0, n\2, binomial(n, k)*b(k)*b(n-k))
A203577(n)=hat(A000108, n) \\ where A000108(n)=(2*n)!/n!/(n+1)! \\ - M. F. Hasler, Jan 13 2012
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CROSSREFS
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Cf. A203576, A000108, A014330 (exponential convolution).
Sequence in context: A051770 A209110 A032181 * A081073 A002831 A114053
Adjacent sequences: A203574 A203575 A203576 * A203578 A203579 A203580
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Jan 13 2012
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STATUS
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approved
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