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A114277
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Sum of the lengths of the second ascents in all Dyck paths of semilength n+2.
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4
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1, 5, 19, 67, 232, 804, 2806, 9878, 35072, 125512, 452388, 1641028, 5986993, 21954973, 80884423, 299233543, 1111219333, 4140813373, 15478839553, 58028869153, 218123355523, 821908275547, 3104046382351, 11747506651599
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also number of Dyck paths of semilength n+4 having length of second ascent equal to three. Example: a(1)=5 because we have UD(UUU)DUDDD, UD(UUU)DDUDD, UD(UUU)DDDUD, UUD(UUU)DDDD and UUDD(UUU)DDD (second ascents shown between parentheses). Partial sums of A002057. Column 3 of A114276. a(n)=absolute value of A104496(n+3).
Also number of Dyck paths of semilength n+3 that do not start with a pyramid (a pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis; here U=(1,1) and D=(1,-1); this definition differs from the one in A091866). Equivalently, a(n)=A127156(n+3,0). Example: a(1)=5 because we have UUDUDDUD, UUDUDUDD, UUUDUDDD, UUDUUDDD and UUUDDUDD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2007
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FORMULA
| a(n)=4sum(binomial(2j+3, j)/(j+4), j=0..n). G.f.=C^4/(1-z), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
a(n)=c(n+3)-[c(0)+c(1)+...c(n+2)], where c(k)=binomial(2k,k)/(k+1) is a Catalan number (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2007
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EXAMPLE
| a(3)=5 because the total length of the second ascents in UD(U)DUD, UD(UU)DD, UUDD(U)D, UUD(U)DD and UUUDDD (shown between parentheses) is 5.
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MAPLE
| a:=n->4*sum(binomial(2*j+3, j)/(j+4), j=0..n): seq(a(n), n=0..28);
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CROSSREFS
| Cf. A002057, A114276, A104496.
Cf. A127156.
Sequence in context: A121525 A163872 A035344 * A104496 A001435 A092492
Adjacent sequences: A114274 A114275 A114276 * A114278 A114279 A114280
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 20 2005
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2007
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