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 1-3 of 3 results.

A104978 Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.

Original entry on oeis.org

1, 1, 1, 2, 5, 3, 5, 21, 28, 12, 14, 84, 180, 165, 55, 42, 330, 990, 1430, 1001, 273, 132, 1287, 5005, 10010, 10920, 6188, 1428, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263, 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675
Offset: 0

Views

Author

Paul D. Hanna, Mar 30 2005

Keywords

Examples

			The triangle T(n, k) begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     5,      3;
  [3]    5,    21,     28,     12;
  [4]   14,    84,    180,    165,     55;
  [5]   42,   330,    990,   1430,   1001,    273;
  [6]  132,  1287,   5005,  10010,  10920,   6188,   1428;
  [7]  429,  5005,  24024,  61880,  92820,  81396,  38760,   7752;
  [8] 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263;
  ...
The array A(n, k) begins:
  [0]   1,    1,      3,      12,       55,       273,       1428, ...  [A001764]
  [1]   1,    5,     28,     165,     1001,      6188,      38760, ...  [A025174]
  [2]   2,   21,    180,    1430,    10920,     81396,     596904, ...  [A383450]
  [3]   5,   84,    990,   10010,    92820,    813960,    6864396, ...  [A383451]
  [4]  14,  330,   5005,   61880,   678300,   6864396,   65615550, ...
  [5]  42, 1287,  24024,  352716,  4476780,  51482970,  551170620, ...
  [6] 132, 5005, 111384, 1899240, 27457584, 354323970, 4206302100, ...
  [A000108]  |  [A074922][A383452]
         [A002054]

Links

Crossrefs

Columns of array: A000108, A002054, A074922, A383452.
Rows of array: A001764, A025174, A383450, A383451.
Cf. A001002 (antidiagonal sums), A001764 (semidiagonal sums), A027307 (row sums), A104979, A383439 (central terms).

Programs

  • Magma
    [Binomial(2*n+k, n+2*k)*Binomial(n+2*k, k)/(n+k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 08 2021
  • Maple
    From Peter Luschny, May 04 2025:  (Start)
T := (n, k) -> (k + 2*n)!/(k!*(n - k)!*(n + k + 1)!):
seq(print(seq(T(n, k), k = 0..n)), n = 0..10);
# Alternatively the array:
A := (n, k) -> (3*k + 2*n)!/(k!*n!*(n + 2*k + 1)!);
for n from 0 to 8 do seq(A(n, k), k = 0..7) od;  (End)
  • Mathematica
    T[n_, k_]:= Binomial[2n+k, n+2k]*Binomial[n+2k, k]/(n+k+1);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jan 27 2019 *)
  • PARI
    T(n,k) = local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1,n,A=1+x*A^2+x*y*A^3); polcoeff(polcoeff(A,n,x),k,y)
for(n=0, 10, for(k=0, n, print1(T(n,k),", ")); print(""))
  • PARI
    Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D,y)); D
T(n,k)=local(A=1); A=1+sum(m=1, n+1, x^m/y^(m+1) * Dy(m-1, (y^2+y^3)^m/m!)) +x*O(x^n)+y*O(y^k); polcoeff(polcoeff(A, n,x),k,y)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Jun 22 2012
  • PARI
    x='x; y='y; z='z; Fxyz = 1 - z + x*z^2 + x*y*z^3;
seq(N) = {
  my(z0 = 1 + O((x*y)^N), z1 = 0);
  for (k = 1, N^2,
    z1 = z0 - subst(Fxyz, z, z0)/subst(deriv(Fxyz, z), z, z0);
    if (z0 == z1, break()); z0 = z1);
  vector(N, n, Vecrev(polcoeff(z0, n-1, 'x)));
};
concat(seq(9)) \\ Gheorghe Coserea, Nov 30 2016
  • Sage
    flatten([[binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 08 2021

Formula

T(n, k) = binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1).
G.f.: A(x, y) = Sum_{n>=0} x^n/y^(n+1) * d^(n-1)/dy^(n-1) (y^2 + y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
G.f. of row n: 1/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
A(n, k) = T(n + k, k) = (3*k + 2*n)! / (k!*n!*(n + 2*k + 1)!). - Peter Luschny, May 04 2025

A274404 Number T(n,k) of modified skew Dyck paths of semilength n with exactly k anti-down steps; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 6, 42, 28, 3, 132, 120, 28, 1, 429, 495, 180, 20, 1430, 2002, 990, 195, 10, 4862, 8008, 5005, 1430, 165, 4, 16796, 31824, 24024, 9009, 1650, 117, 1, 58786, 125970, 111384, 51688, 13013, 1617, 70, 208012, 497420, 503880, 278460, 89180, 16016, 1386, 35
Offset: 0

Views

Author

Alois P. Heinz, Jun 20 2016

Keywords

Comments

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

Examples

			              /\
              \ \
T(3,1) = 1:   /  \
.
Triangle T(n,k) begins:
:     1;
:     1;
:     2;
:     5,     1;
:    14,     6;
:    42,    28,     3;
:   132,   120,    28,    1;
:   429,   495,   180,   20;
:  1430,  2002,   990,  195,   10;
:  4862,  8008,  5005, 1430,  165,   4;
: 16796, 31824, 24024, 9009, 1650, 117, 1;

Links

Crossrefs

Columns k=0-3 give: A000108, A002694(n-1), A074922(n-2), A232224(n-3).
Row sums give A230823.
Last elements of rows give A092392(n-1) for n>0.
Cf. A083920, A274405.

Programs

  • Maple
    b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,
      `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)+
      `if`(t<>1 and x>0, b(x-1, y+1, 2, n-1)*z, 0)+
      `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(0$3, 2*n)):
seq(T(n), n=0..14);
  • Mathematica
    b[x_, y_, t_, n_] := b[x, y, t, n] = Expand[If[y > n, 0,
     If[n == y, If[t == 2, 0, 1], b[x + 1, y + 1, 0, n - 1] +
     If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1] z, 0] +
     If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]]];
T[n_] := CoefficientList[b[0, 0, 0, 2n], z];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

Formula

Sum_{k>0} k * T(n,k) = A274405(n).

A232224 Number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 3 simple intersections.

Original entry on oeis.org

0, 0, 0, 1, 20, 195, 1430, 9009, 51688, 278460, 1434120, 7141530, 34648856, 164663785, 769491450, 3546222225, 16152872400, 72846725160, 325722299760, 1445598337950, 6373942543800, 27942072562950, 121863923024844, 529043313674106, 2287209524819120
Offset: 0

Views

Author

N. J. A. Sloane, Nov 22 2013

Keywords

Links

Crossrefs

Cf. A002694, A074922, A274404.

Programs

  • Mathematica
    CoefficientList[Series[(1 - Sqrt[1 - 4 x^2])^6 ((1 - x^2) Sqrt[1 - 4 x^2] + 7 x^2 - 26 x^4)/(64 x^6 Sqrt[1 - 4 x^2]^5), {x, 0, 48}], x^2] (* Michael De Vlieger, Sep 30 2015 *)
  • PARI
    lista(nn) = {np = 2*nn+2; default(seriesprecision, np); pol = (1-sqrt(1-4*x^2))^6*((1-x^2)*sqrt(1-4*x^2)+7*x^2-26*x^4)/(64*x^6*sqrt(1-4*x^2)^5) + O(x^(np)); forstep (n=0, 2*nn, 2, print1(polcoeff(pol, n), ", "););} \\ Michel Marcus, Sep 30 2015
  • PARI
    x='x+O('x^33); concat([0,0,0],Vec((1-sqrt(1-4*x))^6*((1-x)*sqrt(1-4*x)+7*x-26*x^2) / (64*x^3*sqrt(1-4*x)^5))) \\ Joerg Arndt, Sep 30 2015

Formula

Pilaud-Rue give an explicit g.f.
a(n) = [x^(2n)] (1-sqrt(1-4*x^2))^6*((1-x^2)*sqrt(1-4*x^2)+7*x^2-26*x^4) / (64*x^6*sqrt(1-4*x^2)^5). - Michel Marcus, Sep 30 2015

Extensions

Corrected initial terms and more terms from Lars Blomberg, Sep 30 2015
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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