A108451 -id:A108451 - cp's OEIS frontend

A108453 Number of pyramids of the first kind 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 of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).

1, 5, 29, 201, 1561, 13005, 113525, 1024593, 9482097, 89488213, 857952525, 8332513689, 81805985033, 810551503005, 8094740040677, 81395917522849, 823405075135457, 8374004486010277, 85567502052729597, 878066090712156521

Offset: 1

Emeric Deutsch, Jun 11 2005

nonn

A108453(n)=sum(k*A108451(k),k=1..n) (for example, A108453(3)=1*16+2*5+3*1=29). A108453(n)=(1/2)*A108450(n). A108453 = partial sums of A032349.

			a(2)=5 because in the A027307(2)=10 paths we have altogether 5 pyramids of the first kind (shown between parentheses): (ud)(ud), (ud)Udd, (uudd), uUddd, Udd(ud), UddUdd, Ududd, UdUddd, Uuddd, UUdddd.

Vaclav Kotesovec, Table of n, a(n) for n = 1..160
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A108450, A108451, A032349.

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=z*A^2/(1-z): gser:=series(g,z=0,25): seq(coeff(gser,z^n),n=1..22);

{a(n)=local(y=x); for(i=1, n, y=x*(1+y-x*y)^2/((1-x)*(1-y+x*y)^2) + (O(x^n))^3); polcoeff(y, n)}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014

G.f.=zA^2/(1-z), where A=1+zA^2+zA^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).
G.f. y(x) satisfies: x*(1+y-x*y)^2 = (1-x)*y*(1-y+x*y)^2. - Vaclav Kotesovec, Mar 17 2014
a(n) ~ sqrt(23*sqrt(5)-15) * (11+5*sqrt(5))^n / (11* sqrt(5*Pi) * n^(3/2) * 2^(n+1/2)). - Vaclav Kotesovec, Mar 17 2014
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

A108452 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 no pyramids of the first kind (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).

Original entry on oeis.org

1, 1, 6, 44, 344, 2856, 24816, 223016, 2056256, 19344472, 184956240, 1792088296, 17558218048, 173659691928, 1731556718224, 17387182158184, 175670235597120, 1784561125349464, 18216639085961552, 186762117058304104

Offset: 0

Emeric Deutsch, Jun 11 2005

nonn

Also 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 no pyramids of the second kind (a pyramid of the second kind is a sequence U^pd^(2p) for some positive integer p, starting at the x-axis). Column 0 of A108451.

			a(2)=6 because the paths uUddd, UddUdd, Ududd, UdUddd, Uuddd and UUdddd have no pyramids of the first kind.

Vaclav Kotesovec, Table of n, a(n) for n = 0..100
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A108451, A108449, A108445.

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=(1-z)/(1-z*(1-z)*A*(1+A)): gser:=series(g,z=0,24): 1,seq(coeff(gser,z^n),n=1..21);

{a(n)=local(y=1+x); for(i=1, n, y = -(-1 + 3*x - 3*x^2 + x^3 - 3*x^2*y + 2*x^3*y - 3*x*y^2 + 4*x^2*y^2 - 2*x^3*y^2 + x^4*y^2 - x*y^3 + 5*x^2*y^3 - 5*x^3*y^3 + 2*x^4*y^3) + (O(x^n))^4); polcoeff(y, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 18 2014

G.f.: (1-z)/[1-z(1-z)A(1+A)], where A=1+zA^2+zA^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).
From Vaclav Kotesovec, Mar 18 2014: (Start)
G.f. y(x) satisfies: -1 + 3*x - 3*x^2 + x^3 + y - 3*x^2*y + 2*x^3*y - 3*x*y^2 + 4*x^2*y^2 - 2*x^3*y^2 + x^4*y^2 - x*y^3 + 5*x^2*y^3 - 5*x^3*y^3 + 2*x^4*y^3 = 0
a(n) ~ (11+5*sqrt(5))^n * sqrt(247+603/sqrt(5)) / (5*sqrt(Pi)*n^(3/2) *2^(n+7/2))
Recurrence: n*(2*n + 1)*(6050*n^7 - 126115*n^6 + 1112432*n^5 - 5378320*n^4 + 15373805*n^3 - 25927435*n^2 + 23799813*n - 9117270)*a(n) = (193600*n^9 - 4138530*n^8 + 37940769*n^7 - 194878383*n^6 + 614482575*n^5 - 1224753180*n^4 + 1530842816*n^3 - 1150685847*n^2 + 475947900*n - 86751000)*a(n-1) - 2*(356950*n^9 - 7743285*n^8 + 72449748*n^7 - 382786506*n^6 + 1254763140*n^5 - 2635287165*n^4 + 3523007792*n^3 - 2857685139*n^2 + 1247080365*n - 211094100)*a(n-2) + (629200*n^9 - 13618110*n^8 + 127285773*n^7 - 672901416*n^6 + 2211415230*n^5 - 4666850055*n^4 + 6281980307*n^3 - 5134608429*n^2 + 2249815860*n - 375921000)*a(n-3) - (205700*n^9 - 4402860*n^8 + 40747203*n^7 - 213640971*n^6 + 697768275*n^5 - 1466844360*n^4 + 1971190342*n^3 - 1610202339*n^2 + 703447650*n - 115668000)*a(n-4) - 2*(n-5)*(2*n - 9)*(6050*n^7 - 83765*n^6 + 482792*n^5 - 1496135*n^4 + 2674295*n^3 - 2716295*n^2 + 1400898*n - 257040)*a(n-5)
(End)
D-finite with recurrence +n*(2*n+1)*(72425*n-317734)*a(n) +(-3140100*n^3+18675553*n^2-20491436*n+6673146)*a(n-1) +(22916600*n^3-190703953*n^2+432061605*n-302985732)*a(n-2) +2*(-37979850*n^3+409247558*n^2-1317355900*n+1324935945)*a(n-3) +3*(41724600*n^3-547102003*n^2+2263591341*n-2982348982)*a(n-4) +3*(-36023800*n^3+545643269*n^2-2684061391*n+4314486328)*a(n-5) +(46638250*n^3-790948395*n^2+4390868696*n-7976355570)*a(n-6) +(-6636700*n^3+127715416*n^2-812847607*n+1708833588)*a(n-7) -2*(266400*n-1297177)*(2*n-15)*(n-8)*a(n-8)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

A108453 Number of pyramids of the first kind 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 of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).

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