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A077587
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C(n+1)+nC(n) where C = A000108 (Catalan numbers).
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1
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1, 3, 9, 29, 98, 342, 1221, 4433, 16302, 60554, 226746, 854658, 3239044, 12332140, 47137005, 180780345, 695367510, 2681600130, 10364759790, 40142121030, 155748675420, 605274171060, 2355676013730, 9180275261274, 35819645937228
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of ascents of length 2 starting at an even level in all Dyck paths of semilength n+2. Example: a(1)=3 because all Dyck paths of semilength 3 are UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and UUUDDD, where U=(1,1), D=(1,-1), having alltogether 3 ascents of length 2 that start at an even level (shown between parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 29 2005
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FORMULA
| binomial(2n+1, n+1)-binomial(2n, n+2). a(n)=(3(3n+2)a(n-1)-2(11n-7)a(n-2)+4(2n-5)a(n-3))/(n+2), n>2.
G.f.: A(x)=(1-3x-(1-5x+2x^2)/sqrt(1-4x))/(2x^2) satisfies 0=(x^2+4x-1)+(12x^2-7x+1)A+(4x^3-x^2)A^2.
E.g.f.: A(x) = (1+x)B(x)' where B(x) = e.g.f. of A000108.
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PROG
| (PARI) a(n)=if(n<0, 0, (n^2+6*n+2)*(2*n)!/n!/(n+2)!)
(PARI) a(n)=if(n<0, 0, polcoeff((4+x+1/x-(x+1/x)^2)*(1+x)^(2*n), n)/2)
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CROSSREFS
| Cf. A114462.
Sequence in context: A071740 A081696 A148939 * A001893 A151030 A066331
Adjacent sequences: A077584 A077585 A077586 * A077588 A077589 A077590
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KEYWORD
| nonn,easy
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AUTHOR
| Michael Somos, Nov 09 2002
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