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A026002
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a(n) = T(n,n+2), where T = Delannoy triangle (A008288).
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2
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1, 7, 41, 231, 1289, 7183, 40081, 224143, 1256465, 7059735, 39753273, 224298231, 1267854873, 7178461215, 40704778785, 231128079903, 1314016698401, 7478998203943, 42612705597769, 243025194476551, 1387226559025961, 7924982285747247
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Number of U steps in all lattice paths from (0,0) to (2n,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis (i.e. Schroeder paths). For example, a(2)=7, counting the U's in HH, UDUD, UUDD, UHD, HUD and UDH. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003
Number of UH's in all lattice paths from (0,0) to (2n+2,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis (i.e. Schroeder paths). For example, a(2)=7, counting the UH's, shown between parentheses, in the 22 (=A006318(3)) Schroeder paths of length 6: HHH, HHUD, HUDH, HUDUD, H(UH)D, HUUDD, (UH)DH, (UH)DUD, UUDDH, UUDDUD, (UH)HD, (UH)UDD, UUDHD, UUDUDD, U(UH)DD, UUUDDD, UDHH, UDHUD, UDUDH, UDUDUD, UD(UH)D and UDUUDD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 16 2005
Number of walks from (0,0) to (n+2,n) using steps from {E,N,NE}. [From Shanzhen Gao, May 25 2011]
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FORMULA
| a(n)=(1/n)sum(k binomial(n, k)binomial(n+k, k+1), k=0..n). G.f.: 1/2-1/(2*z)+(1-4*z+z^2)/(2*z*sqrt(1-6*z+z^2)). - Emeric Deutsch, Dec 06 2003
a(n)=sum(k*A110220(n, k), k=0..floor(n/2)). - Emeric Deutsch, Jul 16 2005
a(n)=sum{k=0..n, C(n, k)*C(n+2, k)*2^k}; a(n)=Jacobi_P(n, 2, 0, 3). - Paul Barry, Jan 23 2006
a(n) = (-1)^n*((2*n-1)*LegendreP(n,-3)-LegendreP(n-1,-3))/(2*n+2). - Mark van Hoeij, Oct 31 2011
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MAPLE
| a:=n->(1/n)*sum(k*binomial(n, k)*binomial(n+k, k+1), k=0..n): seq(a(n), n=1..22); (Deutsch)
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CROSSREFS
| Sequence in context: A144635 A097165 A152268 * A173409 A057009 A140480
Adjacent sequences: A025999 A026000 A026001 * A026003 A026004 A026005
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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