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A229955 Triangular array read by rows: 3 dimensional analog of A227997. 0
8, 152, 64, 5056, 2432, 512, 205720, 104000, 29184, 4096, 9305152, 4828544, 1525248, 311296, 32768, 449404224, 236984448, 79898624, 19226624, 3112960, 262144, 22695553536, 12099474432, 4251479040, 1123909632, 221839360, 29884416, 2097152, 1183891745688, 636162156096, 230017430016, 64636047360, 14330265600, 2413559808, 278921216, 16777216 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

T(n,k) is the number of walks on the 3 dimensional grid that start and end at the origin using 2n steps and having exactly k primitive loops. The steps are in the eight directions: (1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1), (-1,1,1), (-1,1,-1), (-1,-1,1), (-1,-1,-1).  A primitive loop is a walk that starts and ends on the origin but does not otherwise touch the origin.

Column 1 is A094059.

Row sums are A002897.

LINKS

Table of n, a(n) for n=1..36.

FORMULA

G.f.: 1/( 1 - y*(1 - 1/A(x)) ) where A(x) is the o.g.f. for A002897.

Generally for such walks in N dimensions:  1/( 1 - y*(1 - 1/B(x)) ) where B(x) = Sum_{n>=0} binomial(2n,n)^N*x^n.

EXAMPLE

8,

152, 64,

5056, 2432, 512,

205720, 104000, 29184, 4096,

9305152, 4828544, 1525248, 311296, 32768,

449404224, 236984448, 79898624, 19226624, 3112960, 262144

MATHEMATICA

nn=6; a=Sum[Binomial[2n, n]^3x^n, {n, 0, nn}]; Map[Select[#, #>0&]&, Drop[CoefficientList[Series[1/(1-y(1-1/a)), {x, 0, nn}], {x, y}], 1]]//Grid

CROSSREFS

Sequence in context: A264642 A300872 A217502 * A249481 A003491 A053606

Adjacent sequences:  A229952 A229953 A229954 * A229956 A229957 A229958

KEYWORD

nonn,tabl,walk

AUTHOR

Geoffrey Critzer, Oct 04 2013

STATUS

approved

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Last modified January 23 13:56 EST 2020. Contains 331171 sequences. (Running on oeis4.)