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 1-5 of 5 results.

A205573 Array M read by antidiagonals in which successive rows evidently converge to A001405 (central binomial coefficients).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 5, 1, 1, 1, 2, 3, 6, 8, 1, 1, 1, 2, 3, 6, 10, 13, 1, 1, 1, 2, 3, 6, 10, 19, 21, 1, 1, 1, 2, 3, 6, 10, 20, 33, 34, 1, 1, 1, 2, 3, 6, 10, 20, 35, 61, 55, 1, 1, 1, 2, 3, 6, 10, 20, 35, 69, 108, 89, 1
Offset: 0

Views

Author

L. Edson Jeffery, Jan 29 2012

Keywords

Comments

CONJECTURE 1. Let M(n,k) (n,k >= 0) denote the entry in row n and column k of the array. For all n, M(n,j) = A001405(j), j=0,...,2*n+1; hence row n of M -> A001405 as n -> infinity.
Taking finite differences of even numbered columns from the top -> down yields triangle A205946 with row sums A000984, central binomial coefficients; while odd numbered columns yield triangle A205945 with row sums A001700. A205946 and A205945 represent the bisection of A191314. - Gary W. Adamson, Feb 01 2012

Examples

			Array begins:
  1, 1, 1, 1, 1,  1,  1,  1,  1,   1,   1,...
  1, 1, 2, 3, 5,  8, 13, 21, 34,  55,  89,...
  1, 1, 2, 3, 6, 10, 19, 33, 61, 108, 197,...
  1, 1, 2, 3, 6, 10, 20, 35, 69, 124, 241,...
  1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 251,...
  1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252,...
  ...
According to Conjecture 2, row n = 3 has g.f. F_3(x) = (1-2*x^2)/(1-x-3*x^2+2*x^3+x^4).

Links

Crossrefs

Cf. A001405, A108299, A115139, A191314.
Cf. also A205945, A205946, A001700, A000984.

Formula

Let N=2*n+3. For each n>0, define the (n+1) X (n+1) tridiagonal unit-primitive matrix (see [Jeffery]) B_n = A_{N,1} = [0,1,0,...,0; 1,0,1,0,...,0; 0,1,0,1,0,...,0; ...; 0,...,0,1,0,1; 0,...,0,1,1], and put B_0 = [1]. Then, for all n, M(n,k)=[(B_n)^k]{n+1,n+1}, k=0,1,..., where X{n+1,n+1} denotes the lower right corner entry of X.
CONJECTURE 2 (Rows of M). Let S(n,i) denote term i in row n of A115139, i=0,...,floor(n/2), and let T(n,j) denote term j in row n of A108299, j=0,...,n. The generating function for row n of M is of the form F_n(x) =sum[i=0,...,floor(n/2) S(n,i)*x^(2*i)]/sum[j=0,...,n T(n,j)*x^j].
CONJECTURE 3 (Columns of M). Let D(m,k) denote term m in column k of A191314, m=0,...,floor(k/2). The generating function for column k of M is of the form G_k(x)=sum[m=0,...,floor(k/2) D(m,k)*x^m]/(1-x).

A205945 Triangle read by rows related to A001700.

Original entry on oeis.org

1, 1, 2, 1, 7, 2, 1, 20, 12, 2, 1, 54, 53, 16, 2, 1, 143, 208, 88, 20, 2, 1, 376, 768, 415, 130, 24, 2, 1, 986, 2734, 1804, 700, 180, 28, 2
Offset: 1

Views

Author

Gary W. Adamson, Feb 01 2012

Keywords

Comments

Row sums = A001700: (1, 3, 10, 35, 126, 462, ...).

Examples

			First few rows of the triangle =
1;
1, 2;
1, 7, 2;
1, 20, 12, 2;
1, 54, 53, 16, 2;
1, 143, 208, 88, 20, 2;
1, 376, 768, 415, 130, 24, 2;
1, 986, 2734, 1804, 700, 180, 28, 2;
...
Row 3 = (1, 7, 2) = row 5 of triangle A191314; and finite differences of column 5 of triangle A205573: (1, 8, 10, ...).

Crossrefs

Cf. A001700, A191314, A205573.

Formula

Bisection of triangle A191314 extracting odd numbered rows. Accessing odd numbered columns of A205573, take finite differences of terms from the top -> down.

A205946 Triangle read by rows related to A000984, central binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 6, 1, 1, 33, 27, 8, 1, 1, 88, 108, 44, 10, 1, 1, 232, 405, 208, 65, 12, 1, 1, 609, 1459, 908, 350, 90, 14, 1
Offset: 0

Views

Author

Gary W. Adamson, Feb 01 2012

Keywords

Examples

			First few rows of the triangle =
1;
1, 1;
1, 4, 1;
1, 12, 6, 1;
1, 33, 27, 8, 1;
1, 88, 108, 44, 10, 1;
1, 232, 405, 208, 65, 12, 1;
1, 609, 1459, 908, 350, 90, 14, 1;
...
Row 2 = (1, 4, 1) = row 4 of triangle A191314.
Row 3 = (1, 12, 6, 1) = finite differences of column 6 of the array shown in A205573: (1, 13, 19, 20)

Crossrefs

Cf. A000984 (row sums), A001405, A191314, A205573, A205945 (companion).

Formula

T(n,k) = A191314(2*n,k).
Take finite differences of even numbered columns of the A205573 array from the top -> down.

A282869 Triangle read by rows: T(n,k) is the number of dispersed Dyck prefixes (i.e., left factors of Motzkin paths with no (1,0) steps at positive heights) of length n and height k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 7, 5, 2, 1, 1, 12, 10, 6, 2, 1, 1, 20, 21, 12, 7, 2, 1, 1, 33, 41, 28, 14, 8, 2, 1, 1, 54, 81, 56, 36, 16, 9, 2, 1, 1, 88, 155, 120, 72, 45, 18, 10, 2, 1, 1, 143, 297, 239, 165, 90, 55, 20, 11, 2, 1, 1, 232, 560, 492, 330, 220, 110, 66, 22, 12, 2, 1, 1, 376, 1054, 974, 715, 440, 286, 132, 78, 24, 13, 2, 1
Offset: 0

Views

Author

Steven Finch, Feb 23 2017

Keywords

Comments

Row n has n+1 entries.

Examples

			Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  2,  1;
  1,  7,  5,  2, 1;
  1, 12, 10,  6, 2, 1;
  1, 20, 21, 12, 7, 2, 1;
  ...
T(4,3) = 2 because we have UUUD and HUUU, where U=(1,1), D=(1,-1), H=(1,0).
T(4,2) = 5 because we have UUDD, UUDU, UDUU, HUUD and HHUU.

Links

Crossrefs

Row sums give A000079.
T(2n,n) gives A283799.
Cf. A000071, A191314.

Programs

  • Maple
    b:= proc(x, y, m) option remember;
      `if`(x=0, z^m, `if`(y>0, b(x-1, y-1, m), 0)+
      `if`(y=0, b(x-1, y, m), 0)+b(x-1, y+1, max(m, y+1)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..16);  # Alois P. Heinz, Mar 13 2017
  • Mathematica
    b[x_, y_, m_] := b[x, y, m] = If[x == 0, z^m, If[y > 0, b[x - 1, y - 1, m], 0] + If[y == 0, b[x - 1, y, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, 0, 0]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, May 12 2017, after Alois P. Heinz *)

Formula

T(n,1) = A000071(n+1), (Fibonacci numbers minus 1).

A191315 Sum of the heights of all dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0) steps at positive heights).

Original entry on oeis.org

0, 0, 1, 2, 6, 11, 27, 50, 115, 216, 481, 913, 1992, 3809, 8192, 15748, 33512, 64685, 136546, 264422, 554686, 1077055, 2248105, 4375221, 9095238, 17735812, 36745504, 71776633, 148288346, 290092160, 597876033, 1171153370, 2408702852, 4723840544, 9697826974, 19038878297
Offset: 0

Views

Author

Emeric Deutsch, May 31 2011

Keywords

Comments

a(n) = Sum_{k>=0} k * A191314(n,k).

Examples

			a(4)=6 because the sum of the heights of the paths HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD is 0+1+1+1+1+2=6; here U=(1,1), H=(1,0), D=(1,-1).

Links

Crossrefs

Cf. A108299, A191314.

Programs

  • Maple
    F[0] := 1: F[1] := 1-z: for k from 2 to 36 do F[k] := sort(expand(F[k-1]-z^2*F[k-2])) end do: G := sum(j*z^(2*j)/(F[j]*F[j+1]), j = 0 .. 34): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 35);
# second Maple program:
b:= proc(x, y, m) option remember;
      `if`(y>x or y<0, 0, `if`(x=0, m, b(x-1, y-1, m)+
      `if`(y=0, b(x-1, y, m), 0)+b(x-1, y+1, max(m, y+1))))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 13 2017
  • Mathematica
    b[x_, y_, m_] := b[x, y, m] = If[y > x || y < 0, 0, If[x == 0, m, b[x - 1, y - 1, m] + If[y == 0, b[x - 1, y, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 16 2017, after Alois P. Heinz *)

Formula

G.f.: G(z) = Sum_{j>=0}(jz^(2j)/(F(j)F(j+1))), where F(k) are polynomials in z defined by F(0)=1, F(1)=1-z, F(k)=F(k-1)-z^2*F(k-2) for k>=2. The coefficients of these polynomials form the triangle A108299.
Showing 1-5 of 5 results.

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