A001263 -id:A001263 - cp's OEIS frontend

A128079 a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1), where A000984 is the central binomial coefficients and A001263 is the Narayana triangle.

1, 3, 13, 69, 411, 2633, 17739, 124029, 892327, 6567285, 49235715, 374841195, 2890994445, 22545855855, 177524073021, 1409591810133, 11275693221519, 90792020672429, 735367765159347, 5987665336600683, 48987680485918149

Offset: 0

Paul D. Hanna, Feb 23 2007

nonn

			Illustrate a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1) by:
a(2) = 1*(1) + 2*(3) + 6*(1) = 13;
a(3) = 1*(1) + 2*(6) + 6*(6) + 20*(1) = 69;
a(4) = 1*(1) + 2*(10)+ 6*(20)+ 20*(10)+ 70*(1) = 411.
The Narayana triangle A001263(n+1,k+1) = C(n,k)*C(n+1,k)/(k+1) begins:
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 10, 20, 10, 1;
1, 15, 50, 50, 15, 1; ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000984, A001263.

Table[Sum[Binomial[2*k,k]*Binomial[n,k]*Binomial[n+1,k]/(k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

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

a(n) = Sum_{k=0..n} C(2k,k)*C(n,k)*C(n+1,k)/(k+1).
Recurrence: (n+1)*(n+2)*a(n) = (7*n^2+11*n+6)*a(n-1) + 3*(7*n^2-19*n+6)*a(n-2) - 27*(n-2)*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3^(2*n+7/2)/(8*Pi*n^2) . - Vaclav Kotesovec, Oct 20 2012
a(n) = ((n+3)^2*A005802(n+1)-(n-3)*(n+1)*A005802(n))/12. - Mark van Hoeij, Nov 12 2023

A132819 A001263 * A127773.

Original entry on oeis.org

1, 1, 3, 1, 9, 6, 1, 18, 36, 10, 1, 30, 120, 100, 15, 1, 45, 300, 500, 225, 21, 1, 63, 630, 1750, 1575, 441, 28, 1, 84, 1176, 4900, 7350, 4116, 784, 36, 1, 108, 2016, 11760, 26460, 24696, 9408, 1296, 45, 1, 135, 3240, 25200, 79380, 111132, 70560, 19440

Offset: 1

Gary W. Adamson, Sep 02 2007

Row sums = A132820: (1, 4, 16, 65, 266, ...).

			First few rows of the triangle are:
  1;
  1,  3;
  1,  9,   6;
  1, 18,  36,  10;
  1, 30, 120, 100,  15;
  1, 45, 300, 500, 225, 21;
  ...

Cf. A001263, A127773, A132820.

A001263 * A127773 as infinite lower triangular matrices.

A136520 a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).

Original entry on oeis.org

1, 1, 3, 13, 44, 146, 530, 1975, 7314, 27262, 102802, 390138, 1486064, 5682756, 21812436, 83976075, 324115550, 1253795510, 4859960402, 18871869302, 73398851448, 285882923196, 1114943553308, 4353426835238, 17016813133124, 66581653586476, 260750813149140, 1022023318047220

Offset: 1

Gary W. Adamson, Jan 02 2008

Narayana transform of A027656.

			a(4) = 13 = (1, 6, 6, 1) dot (1, 0, 2, 0) = (1 + 0 + 12 + 0).
Triangle A001263(n,k) * A027656(k+1) and the rows sums:
  1;                                              :     1;
  1,  0;                                          :     1;
  1,  0,    2;                                    :     3;
  1,  0,   12,  0;                                :    13;
  1,  0,   40,  0,     3;                         :    44;
  1,  0,  100,  0,    45,  0;                     :   146;
  1,  0,  210,  0,   315,  0,     4;              :   530;
  1,  0,  392,  0,  1470,  0,   112,  0;          :  1975;
  1,  0,  672,  0,  5292,  0,  1344,  0,    5;    :  7314;
  1,  0, 1080,  0, 15876,  0, 10080,  0,  225,  0 : 27262;

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001263, A027656.

A136520:= func< n | (&+[((j+1)/(2*j+1))*Binomial(n,2*j)*Binomial(n-1,2*j): j in [0..Floor((n-1)/2)]]) >;
[A136520(n): n in [1..40]]; // G. C. Greubel, Jul 27 2023

A136520[n_]:= Sum[Binomial[n-1, 2*k]*Binomial[n, 2*k]*((k+1)/(2*k+1)), {k,0,Floor[(n-1)/2]}];
Table[A136520[n], {n, 40}] (* G. C. Greubel, Jul 27 2023 *)

def A136520(n): return sum(((j+1)/(2*j+1))*binomial(n,2*j)*binomial(n-1, 2*j) for j in range((n+1)//2))
[A136520(n) for n in range(1,41)] # G. C. Greubel, Jul 27 2023

a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).

a(n) = Sum_{j=0..floor((n-1)/2)} ((j+1)/(2*j+1))*binomial(n, 2*j) * binomial(n-1, 2*j). - G. C. Greubel, Jul 27 2023

Terms a(11) onward added by G. C. Greubel, Jul 27 2023

A136534 A001263 * A128064 (unsigned).

Original entry on oeis.org

1, 2, 2, 4, 8, 3, 7, 24, 21, 4, 11, 60, 90, 44, 5, 16, 130, 300, 260, 80, 6, 22, 2252, 840, 1120, 630, 132, 7, 29, 448, 2058, 3920, 3430, 1344, 203, 8, 37, 744, 4536, 11760, 14700, 9072, 2604, 296, 9, 46, 1170, 9180, 31248, 52920, 46872, 21420, 4680, 414, 10

Offset: 1

Gary W. Adamson, Jan 04 2008

Row sums = A001791: (1, 4, 15, 56, 210, ...).

Left column = A000124: (1, 2, 4, 7, 11, 16, 22, ...).

			First few rows of the triangle:
   1;
   2,   2;
   4,   8,   3;
   7,  24,  21,    4;
  11,  60,  90,   44,   5;
  16, 130, 300,  260,  80,   6;
  22, 252, 840, 1120, 630, 132, 7;
  ...

Cf. A001263, A128064, A001791, A000124.

A001263 * A128064 (unsigned), A001263 = the Narayana triangle, A128064 = an infinite lower triangular matrix with (1, 2, 3, ...) in the main and subdiagonals.

A136535 A128064 * A001263.

Original entry on oeis.org

1, 1, 2, 1, 7, 3, 1, 15, 21, 4, 1, 26, 76, 46, 5, 1, 40, 200, 250, 85, 6, 1, 57, 435, 925, 645, 141, 7, 1, 77, 833, 2695, 3185, 1421, 217, 8, 1, 100, 1456, 6664, 11956, 9016, 2800, 316, 9, 1, 126, 2376, 14616, 37044, 42336, 22176, 5076, 441, 10

Offset: 1

Gary W. Adamson, Jan 04 2008

Row sums give A076540.

			First few rows of the triangle are:
1;
1, 2;
1, 7, 3;
1, 15, 21, 4;
1, 26, 76, 46, 5;
1, 40, 200, 250, 85, 6;
1, 57, 435, 925, 645, 141, 7;
...

Cf. A001263, A128064, A076540.

T4(n,k) = sum(j=k, n, binomial(n,j)*binomial(j,k)*(-1)^(j-k)*(j+1));
T3(n,k) = binomial(n, k)*binomial(n-1, k-1) - binomial(n, k-1)*binomial(n-1, k);
N=10; matrix(N, N, n, k, T4(n-1,k-1))*matrix(N, N,n,k,T3(n,k)) \\ Michel Marcus, Oct 11 2021

a(18) corrected by Georg Fischer, Oct 10 2021

A143778 Eigentriangle of A001263, the Narayana triangle.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 12, 6, 1, 10, 40, 60, 25, 1, 15, 100, 300, 375, 136, 1, 21, 210, 1050, 2625, 2856, 927, 1, 18, 392, 2940, 12250, 26656, 25956, 7690, 1, 36, 672, 7056, 44100, 15993, 311472, 276840, 75913

Offset: 0

Gary W. Adamson, Aug 31 2008

The Narayana triangle begins:
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 10, 20, 10, 1;
...
An eigentriangle of T is generated by taking the termwise product of (n-1)-th row terms of triangle T (in this case the Narayana triangle A001263); and the eigensequence of T = A102812 = (1, 1, 2, 6, 25, 136, 927,...).
Sum of n-th row terms of triangle A143778 = rightmost term of (n+1)-th row.
Right border of the triangle = the eigensequence of T.
Row sums of the triangle = the eigensequence of T shifted one place to the left: (1, 2, 6, 25, 136,...)
(A102812 * 0^(n-k)) = an infinite lower triangular matrix with A102812 as the main diagonal and the rest zeros.

			Triangle begins:
1;
1, 1;
1, 3, 2;
1, 6, 12, 6;
1, 10, 40, 60, 25;
1, 15, 100, 300, 375, 136;
1, 21, 210, 1050, 2625, 2856, 927;
...
Row 3 = (1, 6, 12, 6) = (1*1, 6*1, 6*2, 1*6) = termwise product of row 3 of the Narayana triangle: (1, 6, 6, 1) and the first 4 terms of the eigensequence of the Narayana triangle = (1, 1, 2, 6).

Cf. A001263, A102812.

Triangle read by rows, A001263 * (A102812 * 0^(n-k)); 0<=k<=n

Apparently for kTom Copeland, Oct 08 2014

A155537 Triangle T(n,k,p,q) = (p^n + q^n)*A001263(n, k) with p=2 and q=1, read by rows.

Original entry on oeis.org

3, 5, 5, 9, 27, 9, 17, 102, 102, 17, 33, 330, 660, 330, 33, 65, 975, 3250, 3250, 975, 65, 129, 2709, 13545, 22575, 13545, 2709, 129, 257, 7196, 50372, 125930, 125930, 50372, 7196, 257, 513, 18468, 172368, 603288, 904932, 603288, 172368, 18468, 513

Offset: 1

Roger L. Bagula, Jan 23 2009

			Triangle begins as:
     3;
     5,     5;
     9,    27,      9;
    17,   102,    102,      17;
    33,   330,    660,     330,      33;
    65,   975,   3250,    3250,     975,      65;
   129,  2709,  13545,   22575,   13545,    2709,     129;
   257,  7196,  50372,  125930,  125930,   50372,    7196,    257;
   513, 18468, 172368,  603288,  904932,  603288,  172368,  18468,   513;
  1025, 46125, 553500, 2583000, 5424300, 5424300, 2583000, 553500, 46125, 1025;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000051, A000108, A001263.

T:= func< n,k,p,q | (p^n + q^n)*Binomial(n-1, k-1)*Binomial(n, k)/(n-k+1) >;
[T(n,k,2,1): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 15 2021

A155537:= (n,k,p,q)-> (p^n + q^n)*binomial(n-1, k-1)*binomial(n, k)/(n-k+1);
seq(seq(A155537(n,k,2,1), k=1..n), n=1..12); # G. C. Greubel, Mar 15 2021

T[n_, k_, p_, q_]:= T[n,k,p,q]= (p^n + q^n)*Binomial[n-1, k-1]*Binomial[n, k]/(n-k+1);
Table[T[n,k,2,1], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 15 2021 *)

def T(n,k,p,q): return (p^n + q^n)*binomial(n-1, k-1)*binomial(n, k)/(n-k+1)
flatten([[T(n,k,2,1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 15 2021

Define T(n,k,p,q) = (p^n + q^n)*binomial(n-1, k-1)*binomial(n, k)/(n-k+1) (A scaled Narayana triangle) for various p and q. When p = 2 and q = 1 this sequence is obtained.
From G. C. Greubel, Mar 15 2021: (Start)
T(n,k,p,q) = T(n,k,q,p) = (p^n + q^n)*A001263(n, k).
T(n,k,2,1) = A000051(n) * A001263(n,k).
Sum_{k=1..n} T(n,k,p,q) = (p^n + q^n)*C(n), where C(n) are the Catalan numbers (A000108). (End)

Edited by G. C. Greubel, Mar 15 2021

A157243 Triangle T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 336, 10, 1, 1, 15, 825, 825, 15, 1, 1, 21, 1716, 197676, 1716, 21, 1, 1, 28, 3185, 512050, 512050, 3185, 28, 1, 1, 36, 5440, 1163800, 294296640, 1163800, 5440, 36, 1, 1, 45, 8721, 2395575, 778076145, 778076145, 2395575, 8721, 45, 1

Offset: 0

Roger L. Bagula, Feb 25 2009

			Triangle begins as:
  1;
  1,  1;
  1,  3,    1;
  1,  6,    6,       1;
  1, 10,  336,      10,         1;
  1, 15,  825,     825,        15,         1;
  1, 21, 1716,  197676,      1716,        21,       1;
  1, 28, 3185,  512050,    512050,      3185,      28,    1;
  1, 36, 5440, 1163800, 294296640,   1163800,    5440,   36,  1;
  1, 45, 8721, 2395575, 778076145, 778076145, 2395575, 8721, 45, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000217, A001263, A157219, A157221.

f:= func< n,k | k le Floor(n/2) select k else n-k >;
A001263:= func< n,k | Binomial(n-1,k-1)*Binomial(n,k)/(n-k+1) >;
[A001263(n*f(n,k)+1, f(n,k)+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022

f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
A001263[n_, k_]:= Binomial[n-1,k-1]*Binomial[n,k]/(n-k+1);
T[n_, k_]:= A001263[n*f[n,k] +1, f[n,k] +1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)

def f(n,k): return k if (k <= (n//2)) else n-k
def A001263(n,k): return binomial(n-1,k-1)*binomial(n,k)/(n-k+1)
flatten([[A001263(n*f(n,k)+1, f(n,k)+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022

T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k.
T(n, n-k) = T(n, k).
T(n, 1) = A000217(n). - G. C. Greubel, Jan 11 2022

Edited by G. C. Greubel, Jan 11 2022

A168391 Triangle T(n, k) = [x^k]( p(n,x) ), where p(n, x) = Sum_{k=1..n} A001263(n,k)*binomial(x+k -1, n-1), read by rows.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 6, 19, 21, 14, 24, 84, 126, 84, 42, 120, 468, 750, 720, 330, 132, 720, 2988, 5496, 5445, 3795, 1287, 429, 5040, 22356, 43120, 50435, 35035, 19019, 5005, 1430, 40320, 186912, 391688, 472472, 398398, 208208, 92092, 19448, 4862

Offset: 1

Roger L. Bagula, Nov 24 2009

The Z-transform of the triangle gives the Narayan triangle (A001263).

Every other polynomial, of p(n, x), has a factor of (1+2*x), just like the higher Sierpinski-Pascal Worpitzky form polynomials.

			Triangle begins as:
      1;
      1,      2;
      2,      5,      5;
      6,     19,     21,     14;
     24,     84,    126,     84,     42;
    120,    468,    750,    720,    330,    132;
    720,   2988,   5496,   5445,   3795,   1287,   429;
   5040,  22356,  43120,  50435,  35035,  19019,  5005,  1430;
  40320, 186912, 391688, 472472, 398398, 208208, 92092, 19448, 4862;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A001263, A001710.

p[n_, x_]:= p[n,x]= ((-1)^(n+1)/(n+1))*Sum[Binomial[n-1, k-1]*Binomial[n+1, k]*Pochhammer[1-k-x, n-1], {k, n}];
A168391[n_]:= CoefficientList[p[x, n], x];
Table[A168391[n], {n,12}]//Flatten (* G. C. Greubel, Mar 28 2022 *)

@CachedFunction
def p(n,x): return ((-1)^(n+1)/(n+1))*sum( binomial(n+1, k)*binomial(n-1, k-1)*rising_factorial(1-k-x, n-1) for k in (1..n) )
def A168391(n,k): return ( p(n,x) ).series(x, n+1).list()[k]
flatten([[A168391(n,k) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Mar 28 2022

T(n, k) = [x^k]( p(n,x) ), where p(n, x) = Sum_{k=1..n} A001263(n,k)*binomial(x+k -1, n-1).
From G. C. Greubel, Mar 28 2022: (Start)
Sum_{k=0..n-1} T(n, k) = A001710(n+1).
T(n, 0) = n!. (End)

Edited by G. C. Greubel, Mar 28 2022

A178655 Triangle which contains the first differences of the Catalan triangle A001263 constructed along rows.

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 2, -2, -1, 1, 5, 0, -5, -1, 1, 9, 10, -10, -9, -1, 1, 14, 35, 0, -35, -14, -1, 1, 20, 84, 70, -70, -84, -20, -1, 1, 27, 168, 294, 0, -294, -168, -27, -1, 1, 35, 300, 840, 588, -588, -840, -300, -35, -1

Offset: 0

Roger L. Bagula, Jun 01 2010

			Triangle begins
  1;
  1,   -1;
  1,    0,   -1;
  1,    2,   -2,   -1;
  1,    5,    0,   -5,   -1;
  1,    9,   10,  -10,   -9,   -1;
  1,   14,   35,    0,  -35,  -14,   -1;
  1,   20,   84,   70,  -70,  -84,  -20,   -1;
  1,   27,  168,  294,    0, -294, -168,  -27,   -1;
  1,   35,  300,  840,  588, -588, -840, -300,  -35,   -1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A001263, A000007 (row sums).

[[n le 0 select 1 else ((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))*Binomial(n, k)^2: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 28 2019

Join[{1}, Table[((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))*Binomial[n, k]^2, {n, 1, 10}, {k, 0, n}]//Flatten] (* G. C. Greubel, Jan 28 2019 *)

{T(n,k) = if(n==0, 1, ((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))* binomial(n, k)^2)};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 28 2019

[1] + [[((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))* binomial(n, k)^2 for k in (0..n)] for n in (1..10)] # G. C. Greubel, Jan 28 2019

T(n,k) = -T(n,n-k), n > 0.

T(n,k) = A001263(n,k+1) - A001263(n,k), n > 0. - R. J. Mathar, Jun 16 2015

cp's OEIS Frontend

A128079 a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1), where A000984 is the central binomial coefficients and A001263 is the Narayana triangle.

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A132819 A001263 * A127773.

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A136520 a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).

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A136534 A001263 * A128064 (unsigned).

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A136535 A128064 * A001263.

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A143778 Eigentriangle of A001263, the Narayana triangle.

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A155537 Triangle T(n,k,p,q) = (p^n + q^n)*A001263(n, k) with p=2 and q=1, read by rows.

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A157243 Triangle T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.

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