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Search: 1, 3, 11, 44, 185, 804
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| A127632 |
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Expansion of 1/(1 - x*c(x) * c(x*c(x))), where c(x) is the g.f. of A000108. |
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+20
8
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| 1, 1, 3, 11, 44, 185, 804, 3579, 16229, 74690, 347984, 1638169, 7780876, 37245028, 179503340, 870374211, 4243141332, 20786340271, 102275718924, 505235129250, 2504876652190, 12459922302900, 62167152967680, 311040862133625
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Row sums of number triangle A127631. Hankel transform appears to be A075845.
Catalan transform of Catalan numbers . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 20 2007
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FORMULA
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a(n) = A127714(n+1,2n+1).
G.f. A(x) satisfies 0 = 1 - A(x) + A(x)^2 * x * c(x) where c(x) is the g.f. of A000108.
G.f.: 2/(1 + sqrt( 2 * sqrt(1 -4*x) - 1)). - Michael Somos May 04 2007
a(n)=Sum_{k, 0<=k<=n}A106566(n,k)*A000108(k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 20 2007
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PROGRAM
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(PARI) {a(n)= if(n<1, n==0, polcoeff( serreverse( x*(1-x)^3*(1-x^3)/(1-x^2)^4 +x*O(x^n) ), n))} /* Michael Somos May 04 2007 */
(PARI) {a(n)= local(A); if(n<1, n==0, A= serreverse( x-x^2 +x*O(x^n) ); polcoeff( 1/(1 - subst(A, x, A)), n))} /* Michael Somos May 04 2007 */
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CROSSREFS
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Cf. A127714.
Sequence in context: A132840 A091200 A151105 this_sequence A061706 A167012 A167013
Adjacent sequences: A127629 A127630 A127631 this_sequence A127633 A127634 A127635
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Jan 20 2007, Jan 25 2007
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| A151105 |
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Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, -1, 0), (0, 0, 1), (0, 1, 0), (1, 1, -1), (1, 1, 0)} |
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+20
1
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| 1, 3, 11, 44, 185, 804, 3576, 16179, 74162, 343463, 1603908, 7541122, 35658174, 169421084, 808274631, 3869817810, 18585058316, 89498454123, 432024934133, 2089915753028, 10129261019930, 49178146380312, 239133208092212, 1164443575855438, 5677449527001260, 27713829071829689, 135427236840563801
(list; graph; listen)
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OFFSET
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0,2
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LINKS
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A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.
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MATHEMATICA
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aux[i_Integer, j_Integer, k_Integer, n_Integer] := Which[Min[i, j, k, n] < 0 || Max[i, j, k] > n, 0, n == 0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1 + i, -1 + j, k, -1 + n] + aux[-1 + i, -1 + j, 1 + k, -1 + n] + aux[i, -1 + j, k, -1 + n] + aux[i, j, -1 + k, -1 + n] + aux[1 + i, 1 + j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]
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CROSSREFS
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Sequence in context: A113174 A132840 A091200 this_sequence A127632 A061706 A167012
Adjacent sequences: A151102 A151103 A151104 this_sequence A151106 A151107 A151108
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KEYWORD
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nonn,walk
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AUTHOR
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Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008
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