A108442 Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and having only u steps among the steps leading to the first d step.
1, 1, 3, 15, 97, 721, 5827, 49759, 441729, 4035937, 37702723, 358474735, 3457592161, 33748593841, 332730216579, 3308635650495, 33145196426753, 334193815799233, 3388807714823043, 34537227997917391, 353578650475659617, 3634495706671023505, 37496621681376849219, 388135791657414454815
Offset: 0
Keywords
Examples
a(2)=3 because we have udud, udUdd and uudd.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..955
- Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
Programs
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Maple
A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: gser:=series(1/(1-z*A),z=0,30): 1,seq(coeff(gser,z^n),n=1..25);
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Mathematica
Flatten[{1,Table[Sum[k*Sum[Binomial[2*n-k, i]*Binomial[3*n-2*k-i-1, 2*n-k-1], {i, 0, n-k}]/(2*n-k), {k, 1, n}],{n,1,20}]}] (* Vaclav Kotesovec, Mar 17 2014, after Vladimir Kruchinin *) -
Maxima
a(n):=if n=0 then 1 else sum((k*sum(binomial(2*n-k,i)*binomial(3*n-2*k-i-1,2*n-k-1),i,0,n-k))/(2*n-k),k,1,n); /* Vladimir Kruchinin, Oct 23 2011 */
Formula
G.f.: 1/(1-z*A), where A = 1 + z*A^2 + z*A^3 = (2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
a(n) = Sum_{k=1..n} (k*(Sum_{i=0..n-k} binomial(2*n-k, i)*binomial(3*n-2*k-i-1, 2*n-k-1))/(2*n-k)), n > 0, a(0)=1. - Vladimir Kruchinin, Oct 23 2011
G.f. y(x) satisfies: (3+x)*y*(1-y) + (1+x^2)*y^3 = 1. - Vaclav Kotesovec, Mar 17 2014
a(n) ~ (11+5*sqrt(5))^n / (5^(5/4) * sqrt(Pi) * n^(3/2) * 2^(n+1)). - Vaclav Kotesovec, Mar 17 2014
D-finite with recurrence (2*n-1)*(n-1)*a(n) +6*(n^2-10*n+13)*a(n-1) +(-310*n^2+1869*n-2759)*a(n-2) +48*(-n+3)*a(n-3) +(-310*n^2+1851*n-2705)*a(n-4) +6*(-n^2+2*n+11)*a(n-5) +(n-5)*(2*n-11)*a(n-6)=0. - R. J. Mathar, Jul 26 2022