cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A108449 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 (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).

Original entry on oeis.org

1, 0, 4, 32, 252, 2112, 18484, 166976, 1545548, 14583808, 139774180, 1356966240, 13316740764, 131890671680, 1316627340564, 13234192747648, 133829733962732, 1360586260341248, 13898403178004420, 142578916276009632
Offset: 0

Views

Author

Emeric Deutsch, Jun 11 2005

Keywords

Comments

Column 0 of A108445.

Examples

			a(2)=4 because the paths uUddd, Ududd, UdUddd and Uuddd have no pyramids.

Links

Crossrefs

Cf. A027307, A108445.

Programs

  • Maple
    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-z*(1-z)*A*(1+A)): gser:=series(g,z=0,24): 1,seq(coeff(gser,z^n),n=1..21);
  • PARI
    {a(n)=local(y=1); for(i=1, n, y = -(-1 + 3*x - 3*x^2 + x^3 + 3*x*y - 9*x^2*y + 5*x^3*y - 5*x*y^2 - x^2*y^2 + 5*x^3*y^2 + x^4*y^2 - x*y^3 + 9*x^2*y^3 - 3*x^3*y^3 + 3*x^4*y^3) + (O(x^n))^4); polcoeff(y, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 18 2014

Formula

G.f.=(1-z)/[1+z-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).
G.f. y(x) satisfies: -1 + 3*x - 3*x^2 + x^3 + y + 3*x*y - 9*x^2*y + 5*x^3*y - 5*x*y^2 - x^2*y^2 + 5*x^3*y^2 + x^4*y^2 - x*y^3 + 9*x^2*y^3 - 3*x^3*y^3 + 3*x^4*y^3 = 0. - Vaclav Kotesovec, Mar 18 2014
a(n) ~ (11+5*sqrt(5))^n * sqrt(1738885 + 811683*sqrt(5)) / (961*sqrt(5*Pi) *n^(3/2)*2^(n+3/2)). - Vaclav Kotesovec, Mar 18 2014
Conjecture D-finite with recurrence +n*(2*n+1)*(431*n-2895)*a(n) +2*(-9395*n^3+68622*n^2-64084*n+26109)*a(n-1) +2*(59288*n^3-508196*n^2+1044822*n-574587)*a(n-2) +2*(-94965*n^3+1070605*n^2-3607435*n+3485484)*a(n-3) +4*(29036*n^3-351474*n^2+1402336*n-1970505)*a(n-4) +6*(-6703*n^3+63052*n^2-99178*n-237177)*a(n-5) +6*(1012*n^3-14914*n^2+74580*n-127341)*a(n-6) +6*(1127*n^3-21429*n^2+135199*n-282762
)*a(n-7) +9*(29*n-165)*(2*n-15)*(n-8)*a(n-8)=0. - R. J. Mathar, Jul 26 2022

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).

Original entry on oeis.org

2, 10, 58, 402, 3122, 26010, 227050, 2049186, 18964194, 178976426, 1715905050, 16665027378, 163611970066, 1621103006010, 16189480081354, 162791835045698, 1646810150270914, 16748008972020554, 171135004105459194
Offset: 1

Views

Author

Emeric Deutsch, Jun 11 2005

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

Cf. A027307, A108445, A108453, A032349.

Programs

  • Maple
    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);
  • Mathematica
    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 *)
  • Maxima
    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);
/* Vladimir Kruchinin, Feb 29 2016 */
  • 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)}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014

Formula

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).
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
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
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
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

A108451 Triangle read by rows: T(n,k) is 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 have k 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, 1, 6, 3, 1, 44, 16, 5, 1, 344, 116, 30, 7, 1, 2856, 928, 224, 48, 9, 1, 24816, 7856, 1840, 376, 70, 11, 1, 223016, 69264, 15912, 3184, 580, 96, 13, 1, 2056256, 629472, 142592, 28176, 5080, 844, 126, 15, 1, 19344472, 5855472, 1312360, 256992, 46072
Offset: 0

Views

Author

Emeric Deutsch, Jun 11 2005

Keywords

Comments

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 have k 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). Row sums yield A027307. Column 0 yields A108452. Number of pyramids of the first kind in all paths from (0,0) to (3n,0) is given by A108453.

Examples

			T(2,1)=3 because we have (ud)Udd, (uudd) and Udd(ud), the pyramids of the first kind being shown between parentheses.
Triangle begins:
1;
1,1;
6,3,1;
44,16,5,1;

Links

Crossrefs

Cf. A027307, A108452, A108453, A108445.

Programs

  • Maple
    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-t*z-z*(1-z)*A*(1+A)): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 9 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

Formula

G.f.: (1-z)/[1-tz-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).

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

Views

Author

Emeric Deutsch, Jun 11 2005

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A027307, A108451, A108449, A108445.

Programs

  • Maple
    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);
  • PARI
    {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

Formula

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
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