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A106053
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Next-to-central column of triangle in A059317.
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5
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0, 0, 1, 2, 8, 22, 72, 218, 691, 2158, 6833, 21612, 68726, 218892, 699197, 2237450, 7174018, 23038582, 74097134, 238625222, 769407486, 2483532218, 8024499657, 25951580444, 83999410292, 272098963300, 882045339733, 2861184745710, 9286923094550, 30161343633746
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Number of h steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths hhh, hH, Hh, hUD, UhD and UDh we have altogether 8 h-steps. a(n)=Sum(k*A132277(n-1,k),k=0..n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
Number of paths in the right-half-plane from (0,0) to (n-1,1) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because we have hhU, HU, hUh, Uhh, UH, DUU, UDU and UUD. Number of h-steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths from (0,0) to (3,0), namely, hhh, hH, Hh, hUD, UhD and UDh, we have altogether 8 h-steps. a(n)=Sum(k*A132277(n-1,k),k=0..n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
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LINKS
| W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
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FORMULA
| G.f.=(1 - z - z^2 - sqrt((1+z-z^2)(1-3z-z^2)))/[2sqrt((1+z-z^2)(1-3z-z^2))]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
G.f.=(1-z-z^2)/[2*sqrt((1+z-z^2)(1-3z-z^2))] - 1/2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
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MAPLE
| g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2)): gser:=series(g, z=0, 33); seq(coeff(gser, z, n), n=0..29); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
g:=((1-z-z^2)*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2))-1/2: gser:=series(g, z=0, 33): seq(coeff(gser, z, n), n=0..30); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
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CROSSREFS
| Cf. A059317, A128720, A132277.
Sequence in context: A137103 A089586 A045695 * A121135 A183410 A072929
Adjacent sequences: A106050 A106051 A106052 * A106054 A106055 A106056
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), May 28 2005
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