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Search: id:A108453
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| A108453 |
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Number of pyramids of the first kind in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis). |
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+0
3
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| 1, 5, 29, 201, 1561, 13005, 113525, 1024593, 9482097, 89488213, 857952525, 8332513689, 81805985033, 810551503005, 8094740040677, 81395917522849, 823405075135457, 8374004486010277, 85567502052729597, 878066090712156521
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OFFSET
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1,2
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COMMENT
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A108453(n)=sum(k*A108451(k),k=1..n) (for example, A108453(3)=1*16+2*5+3*1=29). A108453(n)=(1/2)*A108450(n). A108453 =partial sums of A032349.
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REFERENCES
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Problem 10658, American Math. Monthly, 107, 2000, 368-370.
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FORMULA
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G.f.=zA^2/(1-z), where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
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EXAMPLE
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a(2)=5 because in the A027307(2)=10 paths we have alltogether 5 pyramids of the first kind (shown between parentheses): (ud)(ud), (ud)Udd, (uudd), uUddd, Udd(ud), UddUdd, Ududd, UdUddd, Uuddd, UUdddd.
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MAPLE
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A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=z*A^2/(1-z): gser:=series(g, z=0, 25): seq(coeff(gser, z^n), n=1..22);
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CROSSREFS
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Cf. A027307, A108450, A108451, A032349.
Sequence in context: A095000 A086672 A094710 this_sequence A004213 A105277 A103213
Adjacent sequences: A108450 A108451 A108452 this_sequence A108454 A108455 A108456
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 11 2005
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