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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A145599 Triangular array of generalized Narayana numbers: T(n,k) = 5/(n+1)*binomial(n+1,k+4)*binomial(n+1,k-1).

Original entry on oeis.org

1, 5, 5, 15, 35, 15, 35, 140, 140, 35, 70, 420, 720, 420, 70, 126, 1050, 2700, 2700, 1050, 126, 210, 2310, 8250, 12375, 8250, 2310, 210, 330, 4620, 21780, 45375, 45375, 21780, 4620, 330, 495, 8580, 51480, 141570, 196625, 141570, 51480, 8580, 495, 715, 15015
Offset: 4

Views

Author

Peter Bala, Oct 15 2008

Keywords

Comments

T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 4 and which remain in the upper half-plane y >= 0. An example is given in the Example section below.
The current array is the case r = 4 of the generalized Narayana numbers N_r(n,k) := (r + 1)/(n + 1)*binomial(n + 1,k + r)*binomial(n + 1,k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145597 (r = 2) and A145598 (r = 3).

Examples

			Triangle starts
n\k|...1......2......3......4......5......6
===========================================
.4.|...1
.5.|...5......5
.6.|..15.....35.....15
.7.|..35....140....140.....35
.8.|..70....420....720....420.....70
.9.|.126...1050...2700...2700...1050....126
...
T(5,2) = 5: the 5 walks of length 5 from (0,0) to (1,4) are
UUUUR, UUURU, UURUU, URUUU and RUUUU.

Links

Crossrefs

A003519 (row sums), A001263, A145596, A145597, A145598, A145603.

Programs

  • Maple
    with(combinat):
T:= (n,k) -> 5/(n+1)*binomial(n+1,k+4)*binomial(n+1,k-1):
for n from 4 to 13 do
seq(T(n,k),k = 1..n-3);
end do;
  • Mathematica
    Table[5/(n+1) Binomial[n+1,k+4]Binomial[n+1,k-1],{n,4,20},{k,0,n}]/.(0-> Nothing)//Flatten (* Harvey P. Dale, Jan 25 2021 *)

Formula

T(n,k) = 5/(n+1)*binomial(n+1,k+4)*binomial(n+1,k-1) for n >=4 and 1 <= k <= n-3. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n + 2,4). Row sums A003519.
O.g.f. for column k+2: 5/(k + 1) * y^(k+5)/(1 - y)^(k+7) * Jacobi_P(k,5,1,(1 + y)/(1 - y)).
Identities for row polynomials R_n(x) := sum {k = 1..n-3} T(n,k)*x^k:
x^4*R_(n-1)(x) = 5*(n - 1)*(n - 2)*(n - 3)*(n - 4)/((n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)) * sum {k = 0..n} binomial(n + 5,k) * binomial(2n - k,n) * (x - 1)^k;
sum {k = 1..n} (-1)^k*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n-2) = A003519(n)*x^(n-2).
Row generating polynomial R_(n+4)(x) = 5/(n+5)*x*(1-x)^n * Jacobi_P(n,5,5,(1+x)/(1-x)). [From Peter Bala, Oct 31 2008]

A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

Original entry on oeis.org

1, 12, 90, 544, 2907, 14364, 67298, 303600, 1332045, 5722860, 24192090, 100975680, 417225900, 1709984304, 6962078952, 28192122176, 113649492522, 456442180920, 1827459250276, 7297426411968, 29075683360185, 115631433392020, 459124809056550, 1820529677650320, 7210477496434485
Offset: 5

Views

Author

Philippe Deléham, Feb 15 2004

Keywords

Comments

Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.
Number of standard tableaux of shape (n+6,n-5). - Emeric Deutsch, May 30 2004

Links

Crossrefs

Cf. A001622, A009766, A039598, A059365, A214292.
Cf. A000108, A002057, A003517, A003518, A003519.

Programs

  • Mathematica
    Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n,5,50}] (* G. C. Greubel, Feb 07 2017 *)
  • PARI
    for(n=5,50, print1(12*binomial(2*n+1,n-5)/(n+7), ", ")) \\ G. C. Greubel, Feb 07 2017

Formula

a(n) = A039598(n, 5) = A033184(n+7, 12).
G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=11, a(n-6)=(-1)^(n-11)*coeff(charpoly(A,x),x^11). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012
From Karol A. Penson, Nov 21 2016: (Start)
O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.
Recurrence: (-4*(n-5)^2-58*n+80)*a(n+1)-(-n^2-6*n+27)*a(n+2)=0, a(0),a(1),a(2),a(3),a(4)=0,a(5)=1,a(6)=12, n>=5.
Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).
Integral representation as n-th moment of a signed function W(x) on x=(0,4),in Maple notation: a(n+5)=int(x^n*W(x),x=0..4),n=0,1,..., where W(x)=(256/231)*sqrt(4-x)*JacobiP(5, 1/2, 1/2, (1/2)*x-1)*x^(11/2)/Pi and JacobiP are Jacobi polynomials. Note that W(0)=W(4)=0. (End).
From Ilya Gutkovskiy, Nov 21 2016: (Start)
E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.
a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)

Extensions

Missing term 113649492522 inserted by Ilya Gutkovskiy, Dec 07 2016

A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 4, 1, 14, 28, 14, 6, 1, 42, 90, 48, 27, 7, 1, 132, 297, 165, 110, 35, 9, 1, 429, 1001, 572, 429, 154, 54, 10, 1, 1430, 3432, 2002, 1638, 637, 273, 65, 12, 1, 4862, 11934, 7072, 6188, 2548, 1260, 350, 90, 13, 1, 16796, 41990, 25194, 23256, 9996, 5508, 1700, 544, 104, 15, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 01 2014

Keywords

Comments

Row sums are A109262(n+1).

Examples

			Triangle begins:
    1;
    1,   1;
    2,   3,   1;
    5,   9,   4,   1;
   14,  28,  14,   6,  1;
   42,  90,  48,  27,  7, 1;
  132, 297, 165, 110, 35, 9, 1;
Production matrix is:
  1...1
  1...2...1
  0...1...1...1
  0...1...1...2...1
  0...0...0...1...1...1
  0...0...0...1...1...2...1
  0...0...0...0...0...1...1...1
  0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...1...1...1
  0...0...0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...0...0...1...1...1
  ...

Links

Crossrefs

Columns: A000108 (k=0), A000245 (k=1), A002057 (k=2), A003517 (k=3), A000588 (k=4), A001392 (k=5), A003519 (k=6), A090749 (k=7), A000590 (k=8).
Cf. A026674, A026726, A032766, A109262.
Cf. A000045, A039599, A106566.

Programs

  • Magma
    F:=Factorial;
A236843:= func< n,k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
[A236843(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2022
  • Mathematica
    T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)
  • PARI
    T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ Andrew Howroyd, Jan 04 2023
  • SageMath
    F=factorial
def A236843(n,k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
flatten([[A236843(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 13 2022

Formula

G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
Sum_{k=0..n} T(n,k) = A109262(n+1).
Sum_{k=0..n} T(n+k,2k) = A026726(n).
Sum_{k=0..n} T(n+1+k,2k+1) = A026674(n+1).
T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - G. C. Greubel, Jun 13 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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