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%I #50 Apr 23 2023 04:18:33
%S 1,6,54,996,22734,577692,15680628,445162392,13055851998,392475442092,
%T 12029082873372,374482032292008,11808861461931492,376406128925067528,
%U 12108063535794336312,392560994063887113744,12814685828476778001726,420836267423433182275404
%N Number of cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
%H Alois P. Heinz, <a href="/A049037/b049037.txt">Table of n, a(n) for n = 0..200</a>
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/polya/flajolet.html">Symmetric Random Walk on n-Dimensional Integer Lattice</a>
%H Steven R. Finch, <a href="/A054474/a054474.txt">Symmetric Random Walk on n-Dimensional Integer Lattice</a> [Cached copy, with permission of the author]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Define a_0, a_1, ... = [ 1, 6, 54, ... ] by 1+Sum b_i x^i = 1/(1-Sum a_i x^i) where b_0, b_1, ... = [ 1, 6, 90, ... ] = A002896.
%F Or, Sum[ a(n) x^(2n), n=1, 2, ...infinity ] = 1-1/Sum[ A002896(n)*x^(2n), n=0, 1, ...infinity ].
%F G.f.: 2-sqrt(1+12*z) /hypergeom([1/8, 3/8], [1], 64/81*z *(1+sqrt(1-36*z))^2 *(2+sqrt(1-36*z))^4 /(1+12*z)^4)/ hypergeom([1/8, 3/8], [1], 64/81*z *(1-sqrt(1-36*z))^2 *(2-sqrt(1-36*z))^4 /(1+12*z)^4). - _Sergey Perepechko_, Jan 30 2011
%F a(n) ~ c * 36^n / n^(3/2), where c = 0.1014559485279103938501072426734... . - _Vaclav Kotesovec_, Sep 13 2014
%F c = 384 * (3 + 2*sqrt(3)) * Pi^(9/2) / (Gamma(1/24)^4 * Gamma(11/24)^4). - _Vaclav Kotesovec_, Apr 23 2023
%e a(5) = 577692 because there are 577692 different walks that start and end at the origin after 2*5=10 steps, avoiding origin at intermediate steps.
%p read transforms; t1 := [ seq(A002896(i),i=1..25) ]; INVERTi(t1);
%p # second Maple program:
%p b:= proc(n) option remember; `if`(n<2, 5*n+1,
%p (2*(2*n-1)*(10*n^2-10*n+3) *b(n-1)
%p -36*(n-1)*(2*n-1)*(2*n-3) *b(n-2)) /n^3)
%p end:
%p g:= proc(n) g(n):= `if` (n<1, -1, -add(g(n-i) *b(i), i=1..n)) end:
%p a:= n-> abs(g(n)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Nov 02 2012
%t (* A002896 : *) b[n_] := b[n] = Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]; max = 32; a[0] = 1; se = Series[ Sum[ a[n] x^(2 n), {n, 1, max}] - 1 + 1/Sum[ b[n]*x^(2 n), {n, 0, max}], {x, 0, max}]; coes = CoefficientList[se, x]; sol = First[ Solve[ Thread[ coes == 0]]]; Table[ a[n], {n, 0, 16}] /. sol (* _Jean-François Alcover_, Dec 20 2011 *)
%t b[n_] := b[n] = If[n < 2, 5*n + 1, (2*(2*n - 1)*(10*n^2 - 10*n + 3)*b[n-1] - 36*(n - 1)*(2*n - 1)*(2*n - 3)*b[n-2]) / n^3];
%t g[n_] := g[n] = If[n < 1, -1, -Sum [g[n - i]*b[i], {i, 1, n}]];
%t a[n_] := Abs[g[n]];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 12 2018, after _Alois P. Heinz_ *)
%Y Invert A002896, A094059.
%Y Column k=3 of A361397.
%K easy,nonn,nice
%O 0,2
%A Alessandro Zinani (alzinani(AT)tin.it)
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