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Search: id:A088594
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| A088594 |
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Number of 3-dimensional lattice paths running from (0,0,0) to (n,n,n), lying in {(x,y,z) : 0<=x<=y<=z} and using the steps (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1). |
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+0
2
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| 1, 4, 44, 788, 18372, 505156, 15553372, 520065572, 18518471492, 692900847812, 26985709712524, 1086313382608436, 44960426477218436, 1905328431907938180, 82405332511166288572, 3627806131038258219076, 162218975410046793174404
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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These are the 3-dimensional analogues of the large Schroeder numbers, A006318.
R(3,n) = 4*A105124(n) for n>0, where A105124 is the three-dimensional small Schroeder numbers. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 19 2005
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REFERENCES
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R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp.
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LINKS
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R. A. Sulanke, Three-dimensional Narayana and Schr\"oder numbers
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FORMULA
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For n => 1, R(3, n) := Sum[2^(k+2)*Sum[2*(-1)^(k-j)*C(3*n+1, k-j)* C(n+j, n)*C(n+j+1, n)*C(n+j+2, n)/(n+1)^2/(n+2), {j, 0, k}], {k, 0, 2*n-2}]. For n => 4, (3n-4)(n+2)(n+1)^2 R(3, n)(t) = (3n-2)(n+1)( 4(1+t+t^2) - 5(1+7t+t^2)n+3(1+7t+t^2)n^2 )R(3, n-1) - (n-2)( -12 +29n -30n^2 +9n^3)(1-t)^4 R(3, n-2) + (3n-1)(n-2)(n-3)(n-4) (1-t)^6 R(3, n-3)
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MAPLE
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1, seq( add( add(2*(-1)^(k-j)*binomial(3*n+1, k-j)* binomial(n+j, n)*binomial(n+j+1, n)*binomial(n+j+2, n)/(n+1)^2/(n+2), j = 0 .. k) *2^(k+2), k = 0 .. 2*n-2), n = 1 ..20 );
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PROGRAM
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(PARI) {alias(C, binomial); R3(n)=if(n==0, 1, sum(k=0, 2*n-2, 2^(k+2)*sum(j=0, k, 2*(-1)^(k-j)*C(3*n+1, k-j)*C(n+j, n)*C(n+j+1, n)*C(n+j+2, n)/(n+1)^2/(n+2))))} (Hanna)
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CROSSREFS
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Cf. A006318, A001003, A087647.
Adjacent sequences: A088591 A088592 A088593 this_sequence A088595 A088596 A088597
Sequence in context: A103870 A056063 A053332 this_sequence A144004 A053333 A137783
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KEYWORD
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nonn
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AUTHOR
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Robert A. Sulanke (sulanke(AT)math.boisestate.edu), Nov 20 2003
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