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.

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A108440 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 having k u=(2,1) steps among the steps leading to the first d step.

Original entry on oeis.org

1, 1, 1, 5, 4, 1, 33, 25, 7, 1, 249, 184, 54, 10, 1, 2033, 1481, 446, 92, 13, 1, 17485, 12620, 3863, 846, 139, 16, 1, 156033, 111889, 34637, 7881, 1411, 195, 19, 1, 1431281, 1021424, 318812, 74492, 14102, 2168, 260, 22, 1, 13412193, 9536113, 2995228
Offset: 0

Views

Author

Emeric Deutsch, Jun 08 2005

Keywords

Examples

			T(2,1)=4 because we have udud, udUdd, uUddd and Uuddd.
Triangle begins:
.1;
.1,1;
.5,4,1;
.33,25,7,1;
.249,184,54,10,1;

Links

Crossrefs

Row sums yield A027307. Column 0 yields A034015.
Cf. A027307, A034015, A108441.

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/(1-t*z*A-z*A^2): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 9 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
# second Maple program:
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
      `if`(x=0, 1, b(x-1, y-1, false)+b(x-1, y+2, t)+
       b(x-2, y+1, t)*`if`(t, z, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(3*n, 0, true)):
seq(T(n), n=0..10);  # Alois P. Heinz, Oct 06 2015
  • Mathematica
    b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0 || y > x, 0, If[x == 0, 1, b[x - 1, y - 1, False] + b[x - 1, y + 2, t] + b[x - 2, y + 1, t]*If[t, z, 1]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][ b[3*n, 0, True]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 29 2016, after Alois P. Heinz *)

Formula

G.f.: G(t, z)=1/(1-tzA-zA^2)-1, 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).

A108444 Number of triple descents (i.e., ddd's) 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).

Original entry on oeis.org

5, 73, 857, 9505, 103341, 1114969, 11996209, 128989249, 1387480981, 14937170089, 160978217225, 1736820843233, 18760031574077, 202856430706617, 2195832009812065, 23792481053343361, 258038743598973477
Offset: 2

Views

Author

Emeric Deutsch, Jun 10 2005

Keywords

Examples

			a(2)=5 because in the ten paths udud, udUdd, uudd, uU(ddd), Uddud, UddUdd, Ududd, UdU(ddd), Uu(ddd) and UU(d[dd)d] (see A027307) we have 5 ddd's (shown between parentheses).

Links

Crossrefs

Cf. A027307, A108443.

Programs

  • Maple
    A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=z*A*(-z*A-2*z*A^2-2+2*A^2)/(1-3*z*A^2-2*z*A): Gser:=series(G,z=0,26): seq(coeff(Gser,z^n),n=2..21);

Formula

a(n) = Sum_{k=1..2n-1} k*A108443(n,k). Example: a(3) = 1*24 + 2*15 + 3*3 + 4*1 = 73.
G.f.: zA(2A^2-2zA^2-zA-2)/(1-2zA-3zA^2), where A=1+zA^2+zA^3 or, equivalently, A=(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).
Recurrence: n*(2*n+1)*(40*n^5 - 100*n^4 - 758*n^3 + 3649*n^2 - 5474*n + 2727)*a(n) = (880*n^7 - 2200*n^6 - 15316*n^5 + 79354*n^4 - 145332*n^3 + 125379*n^2 - 48111*n + 5220)*a(n-1) + (n-3)*(2*n - 5)*(40*n^5 + 100*n^4 - 758*n^3 + 1175*n^2 - 650*n + 84)*a(n-2). - Vaclav Kotesovec, Mar 18 2014
a(n) ~ 5^(3/4) * ((11+5*sqrt(5))/2)^n / (10*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 18 2014

A371700 G.f. satisfies A(x) = 1 + x * A(x)^6 * (1 + A(x)).

Original entry on oeis.org

1, 2, 26, 482, 10450, 247554, 6208970, 162064322, 4356511138, 119788611458, 3353361311738, 95251219926690, 2738421518770546, 79531905952256642, 2329955712706784682, 68770993359030211458, 2043143866891345880898, 61050342965542475675906
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2024

Keywords

Crossrefs

Cf. A006318, A027307, A144097, A260332, A363006.
Cf. A371678.

Programs

  • PARI
    a(n, r=1, t=6, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+1,n)/(6*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(7*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(6*n,k-1) for n > 0.

A108428 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 doubledescents (i.e., dd's).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 1, 9, 23, 23, 9, 1, 1, 16, 76, 156, 156, 76, 16, 1, 1, 25, 190, 650, 1167, 1167, 650, 190, 25, 1, 1, 36, 400, 2045, 5685, 9318, 9318, 5685, 2045, 400, 36, 1, 1, 49, 749, 5341, 21133, 50813, 77947, 77947, 50813, 21133, 5341, 749, 49, 1, 1, 64, 1288
Offset: 0

Views

Author

Emeric Deutsch, Jun 03 2005

Keywords

Comments

Row n contains 2n terms (n > 0).
Row sums yield A027307.
T(n,1) = T(n,2n-2) = n^2*T(n,k) = T(n,2n-k-1) (mirror symmetry).

Examples

			T(2,1)=4 because we have udUdd, uudd, Uddud and Ududd.
Triangle begins:
  1;
  1,  1;
  1,  4,  4,   1;
  1,  9, 23,  23,   9,  1;
  1, 16, 76, 156, 156, 76, 16, 1;
  ...

Links

Crossrefs

Cf. A027307.

Programs

  • Maple
    a:=proc(n,k) if n=0 and k=0 then 1 elif n=0 then 0 else (1/n)*sum(binomial(n,j)*binomial(n,k-j)*binomial(n+j,k+1),j=0..k) fi end: print(1); for n from 1 to 8 do seq(a(n,k),k=0..2*n-1) od; # yields sequence in triangular form

Formula

T(n, k) = (1/n)*Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*binomial(n+j, k+1).
G.f.: G = G(t, z) satisfies t^2*zG^3 - t^2*zG^2 - (1 + z - tz)G + 1 = 0.

A108437 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 having height of the first peak equal to k.

Original entry on oeis.org

1, 1, 2, 5, 2, 1, 10, 28, 13, 11, 3, 1, 66, 196, 90, 89, 34, 18, 4, 1, 498, 1532, 694, 736, 311, 197, 66, 26, 5, 1, 4066, 12804, 5738, 6344, 2800, 1937, 762, 367, 110, 35, 6, 1, 34970, 111964, 49758, 56576, 25560, 18636, 7953, 4263, 1551, 615, 167, 45, 7, 1
Offset: 1

Views

Author

Emeric Deutsch, Jun 04 2005

Keywords

Comments

Row n contains 2n terms. Row sums yield A027307. T(n,1)=A027307(n-1).

Examples

			T(2,3)=2 because we have uUddd and Uuddd.
Triangle begins:
1,1;
2,5,2,1;
10,28,13,11,3,1;
66,196,90,89,34,18,4,1;

Links

Crossrefs

Cf. A027307.

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/(1-t*z*A-t^2*z*A^2)-1: Gserz:=simplify(series(G,z=0,10)): for n from 1 to 9 do P[n]:=sort(coeff(Gserz,z^n)) od: for n from 1 to 9 do seq(coeff(P[n],t^k),k=1..2*n) od; # yields sequence in triangular form

Formula

G.f.=G=G(t, z)=1/(1-tzA-t^2*zA^2)-1, 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).

A108438 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 having abscissa of the first peak equal to k.

Original entry on oeis.org

1, 1, 4, 3, 2, 1, 24, 18, 13, 7, 3, 1, 172, 130, 96, 55, 28, 12, 4, 1, 1360, 1034, 772, 458, 249, 119, 50, 18, 5, 1, 11444, 8738, 6568, 3982, 2244, 1137, 526, 219, 80, 25, 6, 1, 100520, 76994, 58140, 35770, 20624, 10836, 5293, 2383, 981, 365, 119, 33, 7, 1
Offset: 1

Views

Author

Emeric Deutsch, Jun 04 2005

Keywords

Comments

Row n contains 2n terms. Row sums yield A027307. T(n,1)=A032349(n-1).

Examples

			T(2,3) = 2 because we have Uuddd and uUddd.
Triangle begins:
1,1;
4,3,2,1;
24,18,13,7,3,1;
172,130,96,55,28,12,4,1;

Links

Crossrefs

Cf. A027307, 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:=1/(1-t^2*z*A-t*z*A^2)-1: Gserz:=simplify(series(G,z=0,10)): for n from 1 to 8 do P[n]:=sort(coeff(Gserz,z^n)) od: > for n from 1 to 8 do seq(coeff(P[n],t^k),k=1..2*n) od; # yields sequence in triangular form

Formula

G.f.: G = G(t,z) = 1/(1-t^2zA-tzA^2)-1, 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).

A108439 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 having abscissa of first return equal to 3k.

Original entry on oeis.org

2, 4, 6, 20, 12, 34, 132, 60, 68, 238, 996, 396, 340, 476, 1858, 8132, 2988, 2244, 2380, 3716, 15510, 69940, 24396, 16932, 15708, 18580, 31020, 135490, 624132, 209820, 138244, 118524, 122628, 155100, 270980, 1223134, 5725124, 1872396, 1188980
Offset: 1

Views

Author

Emeric Deutsch, Jun 05 2005

Keywords

Comments

Row sums yield A027307. T(n,n) = A108424(n).

Examples

			T(2,1)=4 because we have u(d)ud, u(d)Udd, Ud(d)ud and Ud(d)Udd, the d step of the first return being shown between parentheses.
Triangle begins:
2;
4,6;
20,12,34;
132,60,68,238;
...

Links

Crossrefs

Cf. A027307, A108424, A108435.

Programs

  • Maple
    a:=n->sum(2^(i+1)*binomial(2*n,i)*binomial(n,i+1),i=0..n-1)/n: b:=proc(n) if n=1 then 2 else (n*2^n*binomial(2*n,n)/((2*n-1)*(n+1)))*sum(binomial(n-1,j)^2/2^j/binomial(n+j+1,j),j=0..n-1): fi end: T:=proc(n,k) if k=n then b(n) else b(k)*a(n-k) fi end:for n from 1 to 9 do seq(T(n,k),k=1..n) od; > # yields sequence in triangular form

Formula

G.f.: tzA(z)A(tz)+tzA(z)A^2(tz), 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).
T(n, k) = A108424(k)*A027307(n-k) (there are explicit formulas in A108424 and A027307).

A109157 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 having sum of the heights of its pyramids equal to k (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis; p is the height of the pyramid).

Original entry on oeis.org

1, 0, 1, 1, 4, 0, 2, 2, 2, 32, 8, 8, 4, 5, 5, 4, 252, 64, 84, 24, 28, 12, 14, 12, 8, 2112, 520, 680, 240, 232, 88, 76, 37, 37, 28, 16, 18484, 4480, 5804, 1992, 2012, 776, 656, 264, 206, 106, 94, 64, 32, 166976, 40008, 51592, 17440, 17400, 6776, 5680, 2392, 1768
Offset: 0

Views

Author

Emeric Deutsch, Jun 20 2005

Keywords

Comments

Row n has 2n+1 terms. Row sums yield A027307. Column 0 yields A108449.

Examples

			T(2,3)=2 because we have udUdd and Uddud.
Triangle begins:
1;
0,1,1;
4,0,2,2,2;
32,8,8,4,5,5,4;
252,64,84,24,28,12,14,12,8;

Links

Crossrefs

Cf. A027307, A108449.

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)*(-1+t^2*z)/(z*(-1+z)*(-1+t^2*z)*(-1+t*z)*A*(1+A)+1-2*t*z-2*t^2*z+z+3*t^3*z^2-t^3*z^3): Gser:=simplify(series(G,z=0,10)): P[0]:=1: for n from 1 to 7 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 7 do seq(coeff(t*P[n],t^k),k=1..2*n+1) od; # yields sequence in triangular form

Formula

G.f. = (1-z)(1-tz)(1-t^2*z)/[1-2tz-2t^2*z+z+3t^3*z^2-t^3*z^3-z(1-z)(1-t^2*z)(1-tz)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).

A109158 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 having height of last peak equal to k.

Original entry on oeis.org

1, 1, 2, 4, 3, 1, 10, 20, 18, 12, 5, 1, 66, 132, 122, 92, 54, 24, 7, 1, 498, 996, 930, 732, 478, 264, 118, 40, 9, 1, 4066, 8132, 7634, 6140, 4214, 2552, 1342, 600, 218, 60, 11, 1, 34970, 69940, 65874, 53676, 37910, 24136, 13782, 7016, 3122, 1180, 362, 84, 13, 1
Offset: 1

Views

Author

Emeric Deutsch, Jun 21 2005

Keywords

Comments

Row n has 2n terms. Row sums yield A027307. T(n,1)=A027307(n-1). T(n,2)=2*A027307(n-1) for n>=2.

Examples

			T(2,3)=3 because we have uUddd, UdUddd and Uuddd.
Triangle begins:
1,1;
2,4,3,1;
10,20,18,12,5,1;
66,132,122,92,54,24,7,1;

Links

Crossrefs

Cf. A027307.

Programs

  • Maple
    A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=t*z*(1+t)/(1-t*z-t^2*z-(1+t)*z*A-z*A^2): Gser:=simplify(series(G,z=0,10)): for n from 1 to 8 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 8 do seq(coeff(P[n],t^k),k=1..2*n) od; # yields sequence in triangular form

Formula

G.f.=tz(1+t)/[1-tz-t^2z-(1+t)zA-zA^2], 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).

A137842 Number of paths from (0,0) if n is even, or from (2,1) if n is odd, to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

Original entry on oeis.org

1, 1, 2, 4, 10, 24, 66, 172, 498, 1360, 4066, 11444, 34970, 100520, 312066, 911068, 2862562, 8457504, 26824386, 80006116, 255680170, 768464312, 2471150402, 7474561164, 24161357010, 73473471344, 238552980386, 728745517972
Offset: 0

Views

Author

Paul Barry, Feb 13 2008

Keywords

Comments

Row sums of the inverse of the Riordan array (1/(1+x^2),x(1-x^2)/(1+x^2)).
a(n) is the maximum number of distinct sets that can be obtained as complete parenthesizations of “S_1 union S_2 intersect S_3 union S_4 intersect S_5 union ... S_{n+1}”, where the total of n union and intersection operations alternate, starting with a union, and S_1, S_2, ... , S_{n+1} are sets. - Alexander Burstein, Nov 22 2023

Crossrefs

Cf. A084078. [From R. J. Mathar, Feb 28 2009]

Formula

G.f.: (1+v^2)/(1-v), where v=2*sqrt(x^2+3)*sin(asin(x(x^2+18)/((x^2+3)^(3/2)))/3)/3-x/3; a(2n)=A027307(n); a(2n+1)=A032349(n+1).
Previous Showing 61-70 of 72 results. Next

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