A203577 - OEIS

A203577

Exponential (or binomial) half-convolution of the sequence A000108 (Catalan) with itself.

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

0,3

COMMENTS

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*Catalan(k)*Catalan(n-k), 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_{n>=0} Catalan(n)^2*x^n/(n!)^2 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).

Recurrence: n*(n+1)^2 * (n+2)^2 * (3*n^6 - 39*n^5 + 166*n^4 - 322*n^3 + 316*n^2 - 153*n + 27)*a(n) = 12*(n-1)*n*(n+1)^2 * (3*n^7 - 34*n^6 + 113*n^5 - 121*n^4 - 19*n^3 + 68*n^2 + 17*n - 18)*a(n-1) + 32*(3*n^11 - 45*n^10 + 220*n^9 - 448*n^8 + 173*n^7 + 920*n^6 - 1696*n^5 + 842*n^4 + 580*n^3 - 846*n^2 + 360*n - 54)*a(n-2) - 768*(n-2)^3 * n *(3*n^7 - 34*n^6 + 113*n^5 - 121*n^4 - 19*n^3 + 68*n^2 + 17*n - 18)*a(n-3) + 2048*(n-3)^3 * (n-2)^2 * (3*n^6 - 21*n^5 + 16*n^4 + 12*n^3 + n^2 - 2)*a(n-4). - Vaclav Kotesovec, Feb 25 2014

a(n) ~ 2^(3*n+2)/(Pi*n^3) * (1 + (1+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, Feb 25 2014

EXAMPLE

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.

MATHEMATICA

a[n_] := Sum[ Binomial[n, k]*CatalanNumber[k]*CatalanNumber[n - k], {k, 0, n/2}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 21 2013 *)

PROG

(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

CROSSREFS

Cf. A203576, A000108, A014330 (exponential convolution).