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A108450 Number of pyramids in all 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) (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).

%I A108450 #26 Jul 26 2022 14:20:59
%S A108450 2,10,58,402,3122,26010,227050,2049186,18964194,178976426,1715905050,
%T A108450 16665027378,163611970066,1621103006010,16189480081354,
%U A108450 162791835045698,1646810150270914,16748008972020554,171135004105459194
%N A108450 Number of pyramids in all 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) (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).
%C A108450 A108450(n)=sum(k*A108445(k),k=1..n) (for example, A108450(3)=1*18+2*8+3*8=58). A108450(n)=2*A108453(n). A108450 =2*partial sums of A032349.
%H A108450 Vaclav Kotesovec, <a href="/A108450/b108450.txt">Table of n, a(n) for n = 1..140</a>
%H A108450 Emeric Deutsch, <a href="http://www.jstor.org/stable/2589192">Problem 10658: Another Type of Lattice Path</a>, American Math. Monthly, 107, 2000, 368-370.
%F A108450 G.f.: 2*z*A^2/(1-z), 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).
%F A108450 G.f. y(x) satisfies: y = (2*x*(2+y-x*y)^2)/((1-x)*(2-y+x*y)^2). - _Vaclav Kotesovec_, Mar 17 2014
%F A108450 a(n) ~ (3*sqrt(5)-1) * ((11+5*sqrt(5))/2)^n /(11*5^(1/4)*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 17 2014
%F A108450 a(n) = 2*Sum_{k=0..n-1}(Sum_{i=0..k}(binomial(2*k+2,k-i)* binomial(2*k+i+1,2*k+1))/(k+1)). - _Vladimir Kruchinin_, Feb 29 2016
%F A108450 D-finite with recurrence n*(2*n-1)*a(n) +6*-(n-1)*(5*n-6)*a(n-1) +4*(23*n^2-97*n+111)*a(n-2) +2*(-29*n^2+142*n-174)*a(n-3) -3*(2*n-5)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%e A108450 a(2)=10 because in the A027307(2)=10 paths we have altogether 10 pyramids (shown between parentheses): (ud)(ud), (ud)(Udd), (uudd), uUddd, (Udd)(ud), (Udd)(Udd), Ududd, UdUddd, Uuddd, (UUdddd).
%p A108450 A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=2*z*A^2/(1-z): gser:=series(g,z=0,25): seq(coeff(gser,z^n),n=1..22);
%t A108450 Table[2 Sum[Sum[Binomial[2 k + 2, k - i] Binomial[2 k + i + 1, 2 k + 1], {i, 0, k}]/(k + 1), {k, 0, n - 1}], {n, 19}] (* _Michael De Vlieger_, Feb 29 2016 *)
%o A108450 (PARI) {a(n)=local(y=2*x); for(i=1, n, y=(2*x*(2+y-x*y)^2)/((1-x)*(2-y+x*y)^2) + (O(x^n))^3); polcoeff(y, n)}
%o A108450 for(n=1, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Mar 17 2014
%o A108450 (Maxima)
%o A108450 a(n):=2*sum(sum(binomial(2*k+2,k-i)*binomial(2*k+i+1,2*k+1),i,0,k)/(k+1),k,0,n-1);
%o A108450 /* _Vladimir Kruchinin_, Feb 29 2016 */
%Y A108450 Cf. A027307, A108445, A108453, A032349.
%K A108450 nonn
%O A108450 1,1
%A A108450 _Emeric Deutsch_, Jun 11 2005