A119245 -id:A119245 - cp's OEIS frontend

A119244 Eigenvector of triangle A119245, so that a(n) = Sum_{k=0..floor(n/2)} A119245(n,k)*a(k).

1, 1, 3, 10, 37, 144, 579, 2379, 9927, 41905, 178498, 765876, 3305926, 14342635, 62496000, 273347910, 1199555865, 5279624910, 23298471120, 103057254030, 456833819925, 2028998199360, 9027655451400, 40232167135245

Offset: 0

Paul D. Hanna, May 10 2006

nonn

Self-convolution equals A119243, which is the eigenvector of triangle A118919.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A119245, A119243 (self-convolution).

{a(n)=if(n==0,1,sum(k=0,n\2,a(k)*(4*k+1)*binomial(2*n+1,n-2*k)/(2*n+1)))}

seq(n) = {my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n] = sum(k=0, n\2, a[1+k]*(4*k+1)*binomial(2*n+1,n-2*k))/(2*n+1)); a} \\ Andrew Howroyd, Sep 19 2023

G.f. A(x) satisfies: A(x) = A(-x/(1-4*x))/sqrt(1-4*x).
Eigenvector: a(n) = Sum_{k=0..floor(n/2)} a(k)*(4*k+1)*binomial(2*n+1,n-2*k)/(2*n+1) for n>=0, with a(0)=1.
From Peter Bala, Sep 16 2023: (Start)
It appears that the g.f. A(x) satisfies A(x^2) = 1/(1 + x)*A(x/(1 + x)^2).
If true, then F(x) := A(x^2) satisfies the functional equation F(x^2) = 1/(1 + x^2)*F(x/(1 + x^2)). Compare with Landen's transformation K(x^2) = 1/(1 + x^2)*K(2*x/(1 + x^2)) (for 0 <= x < 1) for the complete elliptic integral of the first kind K(x) = Integral_{t = 0..Pi/2} dt/sqrt(1 - x^2*sin^2(t)). (End)

A118919 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).

Original entry on oeis.org

1, 2, 5, 1, 14, 6, 42, 27, 1, 132, 110, 10, 429, 429, 65, 1, 1430, 1638, 350, 14, 4862, 6188, 1700, 119, 1, 16796, 23256, 7752, 798, 18, 58786, 87210, 33915, 4655, 189, 1, 208012, 326876, 144210, 24794, 1518, 22, 742900, 1225785, 600875, 123970, 10350

Offset: 0

Emeric Deutsch, May 06 2006

Row n contains 1+floor(n/2) terms. Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A003517(n). T(n,2)=A003519(n). Sum(k*T(n,k),k>=0)=A008549(n-1). For both downward and upward crossings, see A118920.
Eigenvector is defined by: A119243(n) = Sum_{k=0..[n\2]} T(n,k)*A119243(k). This triangle is closely related to triangle A119245. - Paul D. Hanna, May 10 2006
Column k is the sum of columns 2k and 2k+1 of A039599. - Philippe Deléham, Nov 11 2008

			T(3,1)=6 because we have ud\dudu,ud\dduu,udud\du,uudd\du,ud\duud and duud\du (the downward crossings of the x-axis are shown by a back-slash \).
Triangle starts:
  1;
  2;
  5,1;
  14,6;
  42,27,1;
  132,110,10;

Cf. A000984, A000108, A003517, A003519, A008549, A118920.

Cf. A119243 (eigenvector), A119245 (variant).

T:=(n,k)->(2*k+1)*binomial(2*n+2,n-2*k)/(n+1): for n from 0 to 13 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

T(n,k)=if(n<2*k || k<0,0,(2*k+1)*binomial(2*n+2,n-2*k)/(n+1)) \\ Paul D. Hanna, May 10 2006

T(n,k)=(2k+1)binomial(2n+2,n-2k)/(n+1). G.f.=G(t,z)=C^2/(1-tz^2*C^4), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

T(n,k)=A039599(n,2k)+A039599(n,2k+1). - Philippe Deléham, Nov 11 2008

A119243 Eigenvector of triangle A118919, so that a(n) = Sum_{k=0..floor(n/2)} A118919(n,k)*a(k).

Original entry on oeis.org

1, 2, 7, 26, 103, 422, 1768, 7520, 32335, 140174, 611530, 2681516, 11807683, 52177166, 231262945, 1027703054, 4577477065, 20429990450, 91348096963, 409110897122, 1834954888618, 8241277167236, 37059369415102

Offset: 0

Paul D. Hanna, May 10 2006

nonn

Equals the self-convolution of A119244, which is the eigenvector of triangle A119245.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A118919, A119244 (A(x)^(1/2)), A119245.

{a(n)=if(n==0,1,sum(k=0,n\2,a(k)*(2*k+1)*binomial(2*n+2,n-2*k)/(n+1)))}

seq(n) = {my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n] = sum(k=0, n\2, a[1+k]*(2*k+1)*binomial(2*n+2,n-2*k))/(n+1)); a} \\ Andrew Howroyd, Sep 19 2023

G.f. A(x) satisfies: A(x) = A(-x/(1-4*x))/(1-4*x).
Eigenvector: a(n) = Sum_{k=0..floor(n/2)} a(k)*(2*k+1)*binomial(2*n+2,n-2*k)/(n+1) for n>=0, with a(0)=1.
It appears that the g.f. A(x) satisfies A(x^2) = 1/(1 + x)^2*A(x/(1 + x)^2). - Peter Bala, Sep 16 2023

A158483 Triangle read by rows: T(n,k) = (4k+3)/(n+2k+2)*binomial(2n,n+2k+1).

Original entry on oeis.org

0, 1, 3, 9, 1, 28, 7, 90, 35, 1, 297, 154, 11, 1001, 637, 77, 1, 3432, 2548, 440, 15, 11934, 9996, 2244, 135, 1, 41990, 38760, 10659, 950, 19, 149226, 149226, 48279, 5775, 209, 1, 534888, 572033, 211508, 31878, 1748, 23, 1931540, 2187185, 904475, 164450

Offset: 0

Peter Bala, Mar 20 2009

This triangle forms a companion to A119245.
Combinatorial interpretations of T(n,k):
1) The number of standard tableaux of shape (n-2*k-1,n+2*k+1).
2) The entries in column k are (with an offset of 2*k+1) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k+2. See [Sunik, Theorem 4].

			Triangle begins
==================================
n\k|.....0.....1.....2.....3.....4
==================================
.0.|.....0
.1.|.....1
.2.|.....3
.3.|.....9.....1
.4.|....28.....7
.5.|....90....35.....1
.6.|...297...154....11
.7.|..1001...637....77.....1
.8.|..3432..2548...440....15
.9.|.11934..9996..2244...135.....1

Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).

Cf. A000245 (column 0), A000588 (column 1), A000589 (column 2), A001700 (row sums), A119245.

with(combinat): T:=(n,k) -> (4k+3)/(n+2k+2)*binomial(2n,n+2k+1): for n from 0 to 13 do seq(T(n,k),k = 0..6); end do;

T(n,k) = (4*k+3)/(n+2*k+2)*binomial(2*n,n+2*k+1).
O.g.f. y*C(y)^3/(1 - x*y^2*C(y)^4) = y + 3*y^2 + (9 + x)*y^3 + (28 + 7*x)*y^4 + ..., where C(x) = [1-(1-4*x)^(1/2)]/(2*x) is the o.g.f. for the Catalan numbers A000108.
Row sums A001700.

cp's OEIS Frontend

A119244 Eigenvector of triangle A119245, so that a(n) = Sum_{k=0..floor(n/2)} A119245(n,k)*a(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A118919 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Formula

A119243 Eigenvector of triangle A118919, so that a(n) = Sum_{k=0..floor(n/2)} A118919(n,k)*a(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A158483 Triangle read by rows: T(n,k) = (4k+3)/(n+2k+2)*binomial(2n,n+2k+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula