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Search: id:A117813
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| A117813 |
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Consider 1-D random walk with jumps up to the third neighbor, i.e. set of possible jumps is {-3,-2,-1,+1,+2,+3}. Sequence gives number of paths of length n ending at origin. |
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+0
1
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| 1, 0, 6, 18, 122, 600, 3450, 18914, 107338, 606816, 3466356, 19852470, 114239642, 659275760, 3815952426, 22138925718, 128718762250, 749773729952, 4374616990332, 25561798008252
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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Recurrence: 36864*(n + 1)*(n + 2)*(n + 3)*a[n] - 3072*(n + 2)*(n + 3)*(97*n + 142)*a[n + 1] - 64*(n + 3)*(4031*n^2 + 17601*n + 19504)*a[n + 2] - (26944*n^3 + 215856*n^2 + 498848*n + 243840)*a[n + 3] + (15912*n^3 + 173328*n^2 + 687072*n + 997512)*a[n + 4] + (1868*n^3 + 28044*n^2 + 143368*n + 249960)*a[n + 5] - 2*(n + 6)*(115*n^2 + 1080*n + 2273)*a[n + 6] - 3*(n + 7)*(3*n + 19)*(3*n + 20)*a[n + 7]=0
ODE for G.f (in Maple notation) x^2*(6*x - 1)^2*(8*x + 1)^2*(2*x + 1)*(8*x^2 - 68*x - 27)*diff(G(x),x$3) + 6*x*(6*x - 1)*(8*x + 1)*(1152*x^5 - 6640*x^4 - 4164*x^3 - 500*x^2 - 3*x + 9)*diff(G(x),x$2) + 6*(110592*x^7 - 390144*x^6 - 122048*x^5 + 11416*x^4 + 10420*x^3 + 820*x^2 + 84*x - 1)*diff(G(x),x) + 24*x*(9216*x^5 - 11520*x^4 - 1136*x^3 + 1562*x^2 + 171*x + 30)*G(x)=0
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MAPLE
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a:=array(0..25, [1, 0, 6, 18, 122, 600, 3450]): for n from 0 to 18 do a[n + 7]:=(36864*(n + 1)*(n + 2)*(n + 3)*a[n] - 3072*(n + 2)*(n + 3)*(97*n + 142)*a[n + 1] - 64*(n + 3)*(4031*n^2 + 17601*n + 19504)*a[n + 2] - (26944*n^3 + 215856*n^2 + 498848*n + 243840)*a[n + 3] + (15912*n^3 + 173328*n^2 + 687072*n + 997512)*a[n + 4] + (1868*n^3 + 28044*n^2 + 143368*n + 249960)*a[n + 5] - 2*(n + 6)*(115*n^2 + 1080*n + 2273)*a[n + 6])/(3*(n + 7)*(3*n + 19)*(3*n + 20)) od;
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CROSSREFS
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Cf. A092765.
Sequence in context: A009576 A009580 A125839 this_sequence A012758 A003496 A009582
Adjacent sequences: A117810 A117811 A117812 this_sequence A117814 A117815 A117816
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KEYWORD
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nonn
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AUTHOR
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Sergey Perepechko (persn(AT)aport.ru), Apr 30 2006
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