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1, 1, 4, 11, 36, 117, 393, 1339, 4630, 16193, 57201, 203799, 731602, 2643903, 9611748, 35130195, 129018798, 475907913, 1762457595, 6550726731, 24428809690, 91377474411, 342763939656, 1289070060903
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = p(n + 1) where p(x) is the unique degree-n polynomial such that p(k) = Catalan(k) for k = 0, 1, ..., n. - Michael Somos, Jan 05 2012
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FORMULA
| G.f.: (sqrt( 1 - 2*x - 3*x^2 ) / (1 + x) - sqrt( 1 - 4*x )) / (2*x^2) = 2 / (sqrt( 1 - 2*x - 3*x^2 ) + (1 + x) * sqrt( 1 - 4*x )). - Michael Somos, Jan 05 2012
a(n) = A000108(n+1) - A005043(n+1).
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EXAMPLE
| 1 + x + 4*x^2 + 11*x^3 + 36*x^4 + 117*x^5 + 393*x^6 + 1339*x^7 + 4630*x^8 + ...
a(2) = 4 since p(x) = (x^2 - x + 2) / 2 interpolates p(0) = 1, p(1) = 1, p(2) = 2, and p(3) = 4. - Michael Somos, Jan 05 2012
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PROG
| (PARI) {a(n) = if( n<0, 0, n++; subst( polinterpolate( vector(n, k, binomial( 2*k - 2, k - 1) / k)), x, n + 1))} /* Michael Somos, Jan 05 2012 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 / (sqrt( 1 - 2*x - 3*x^2 + A) + (1 + x) * sqrt( 1 - 4*x + A)) , n))} /* Michael Somos, Jan 05 2012 */
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CROSSREFS
| Cf. A000108, A005043.
Sequence in context: A149237 A054577 A206687 * A109268 A174993 A149238
Adjacent sequences: A106637 A106638 A106639 * A106641 A106642 A106643
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KEYWORD
| nonn
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AUTHOR
| Philippe DELEHAM, May 26 2005
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