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A151281
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Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (1, 0), (1, 1)}
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6
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1, 2, 6, 16, 48, 136, 408, 1184, 3552, 10432, 31296, 92544, 277632, 824448, 2473344, 7365120, 22095360, 65920000, 197760000, 590790656, 1772371968, 5299916800, 15899750400, 47578857472, 142736572416, 427357700096, 1282073100288, 3840133464064, 11520400392192, 34517383151616, 103552149454848
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 26 2009: (Start)
Image of 2^n under A155761. Binomial transform is A129637. Hankel transform is 2^C(n+1,2).
In general, the g.f. of the reversion of x(1+cx)/(1+ax+bx^2) is given by the continued fraction
x/(1-(a-c)x-(b-ac+c^2)x^2/(1-(a-2c)x-(b-ac+c^2)x^2/(1-(a-2c)x-(b-ac+c^2)x^2/(1-.... (End)
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LINKS
| M. Bousquet-Melou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.
A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.
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FORMULA
| Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 26 2009: (Start)
G.f.: 1/(1-2x-2x^2/(1-2x^2/(1-2x^2/(1-2x^2/(1-2x^2/(1-.... (continued fraction);
G.f.: c(2x^2)/(1-2xc(2x^2))=(sqrt(1-8x^2)+4x-1)/(4x(1-3x));
a(n)=sum{k=0..n, ((k+1)/(n+k+1))*C(n,(n-k)/2)*(1+(-1)^(n-k))*2^((n-k)/2)*2^k};
Reversion of x(1+2x)/(1+4x+6x^2). (End)
a(n)=sum{k=0..n, A120730(n,k)*2^k}. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 01 2009]
a(2n+2)=3*a(2n+1), a(2n+1)=3*a(2n)-2^n*A000108(n) = 3*a(2n)-A151374(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 02 2009]
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MATHEMATICA
| aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]
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CROSSREFS
| Sequence in context: A148443 A148444 A064190 * A045694 A129772 A046721
Adjacent sequences: A151278 A151279 A151280 * A151282 A151283 A151284
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KEYWORD
| nonn,walk
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AUTHOR
| Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008
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