A001263 -id:A001263 - cp's OEIS frontend

A132789 Triangle read by rows: T(n,k) = A007318(n-1, k-1) + A001263(n, k) - 1.

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 25, 13, 1, 1, 19, 59, 59, 19, 1, 1, 26, 119, 194, 119, 26, 1, 1, 34, 216, 524, 524, 216, 34, 1, 1, 43, 363, 1231, 1833, 1231, 363, 43, 1, 1, 53, 575, 2603, 5417, 5417, 2603, 575, 53, 1, 1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869

Offset: 1

Gary W. Adamson, Aug 30 2007

			First few rows of the triangle are:
  1;
  1,  1;
  1,  4,   1;
  1,  8,   8,    1;
  1, 13,  25,   13,     1;
  1, 19,  59,   59,    19,     1;
  1, 26, 119,  194,   119,    26,     1;
  1, 34, 216,  524,   524,   216,    34,    1;
  1, 43, 363, 1231,  1833,  1231,   363,   43,   1;
  1, 53, 575, 2603,  5417,  5417,  2603,  575,  53,  1;
  1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869, 64, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Column k=2 is A034856.
Row sums are A132790.
Cf. A007318, A001263.

<< DiscreteMath`Combinatorica`
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1);
t[n_, m_, 2] := Eulerian[1 + n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

T(n,k)={if(k<=n, binomial(n-1, k-1)*(1 + binomial(n, k-1)/k) - 1, 0)}
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Sep 08 2018

Equals A007318 + A001263 - A000012 as infinite lower triangular matrices.
A symmetrical triangle recursion: let q=4; t(n,m,0)=Binomial[n,m]; t(n,m,1)=Narayana(n,m); t(n,m,2)=Eulerian(n+1,m); t(n,m,q)=t(n,m,g-2)+t(n,m,q-3).
T(n,k) = binomial(n-1, k-1)*(1 + binomial(n, k-1)/k) - 1. - Andrew Howroyd, Sep 08 2018

More terms, Mma program and additional comments from Roger L. Bagula, Apr 20 2010
Edited by N. J. A. Sloane, Apr 21 2010 at the suggestion of R. J. Mathar
Name clarified by Andrew Howroyd, Sep 08 2018

A136536 Triangle read by rows: A001263 * A128064 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 5, 7, 3, 14, 19, 19, 4, 42, 51, 71, 41, 5, 132, 146, 216, 216, 76, 6, 429, 449, 617, 827, 547, 127, 7, 1430, 1457, 1793, 2675, 2675, 1205, 197, 8, 4862, 4897, 5497, 8017, 10369, 7429, 2389, 289, 9, 16796, 16840, 17830, 23770, 34858, 34858, 18226, 4366, 406, 10

Offset: 1

Gary W. Adamson, Jan 04 2008

Row sums = A001791: (1, 4, 15, 56, 210, 792, ...).

Left column = A000108 starting (1, 2, 5, 14, 42, 132, 429, ...).

			First few rows of the triangle:
    1;
    2,   2;
    5,   7,   3;
   14,  19,  19,   4;
   42,  51,  71,  41,   5;
  132, 146, 216, 216,  76,   6;
  429, 449, 617, 827, 547, 127,   7;
  ...

Cf. A000108, A001263, A001791, A128064.

a(46) = 16796 corrected and two more terms from Georg Fischer, May 31 2023

A137940 Triangle read by rows, antidiagonals of an array formed by A000012 * A001263 (transform).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 7, 1, 1, 2, 5, 13, 11, 1, 1, 2, 5, 14, 31, 16, 1, 1, 2, 5, 14, 41, 66, 22, 1, 1, 2, 5, 14, 42, 116, 127, 29, 1, 1, 2, 5, 14, 42, 131, 302, 225, 37, 1, 1, 2, 5, 14, 42, 132, 407, 715, 373, 46, 1, 1, 2, 5, 14, 42, 132, 428, 1205, 1549, 586, 56, 1

Offset: 1

Gary W. Adamson, Feb 24 2008

Rows of the array tend to the Catalan sequence, A000108 starting (1, 2, 5, 14, 42, ...).

			First few rows of the array:
  1, 1, 1,  1,  1, ...
  1, 2, 4,  7, 11, ...
  1, 2, 5, 13, 31, ...
  1, 2, 5, 14, 41, ...
  1, 2, 5, 14, 42, ...
  ...
First few rows of the triangle:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 4,  1;
  1, 2, 5,  7,  1;
  1, 2, 5, 13, 11,   1;
  1, 2, 5, 14, 31,  16,   1;
  1, 2, 5, 14, 41,  66,  22,   1;
  1, 2, 5, 14, 42, 116, 127,  29,   1;
  1, 2, 5, 14, 42, 131, 302, 225,  37,  1;
  1, 2, 5, 14, 42, 132, 407, 715, 373, 46, 1;
  ...

Antonio Bernini, Matteo Cervetti, Luca Ferrari, Einar Steingrimsson, Enumerative combinatorics of intervals in the Dyck pattern poset, arXiv:1910.00299 [math.CO], 2019. See Table 1 p. 4.

Cf. A001263, A000108, A106396.

Antidiagonals of an array formed by A000012 * A001263(transform), as infinite triangular matrices. A000012 = (1; 1,1; 1,1,1; 1,1,1,1; ...), A001263 = the Narayana triangle.

More terms from Alois P. Heinz, Nov 28 2021

A157118 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(nk+1, n-k+1) if k <= n otherwise A001263(n(n-k)+1, k+1) and T(1, k) = 1, read by rows.

Original entry on oeis.org

2, 1, 1, 1, 6, 1, 1, 27, 27, 1, 1, 88, 672, 88, 1, 1, 225, 9150, 9150, 225, 1, 1, 486, 98385, 395352, 98385, 486, 1, 1, 931, 1126951, 11748681, 11748681, 1126951, 931, 1, 1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1, 1, 2673, 201755880, 16093941435, 32251030119, 32251030119, 16093941435, 201755880, 2673, 1

Offset: 0

Roger L. Bagula, Feb 23 2009

			Triangle begins as:
  2;
  1,    1;
  1,    6,        1;
  1,   27,       27,         1;
  1,   88,      672,        88,         1;
  1,  225,     9150,      9150,       225,         1;
  1,  486,    98385,    395352,     98385,       486,        1;
  1,  931,  1126951,  11748681,  11748681,   1126951,      931,    1;
  1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1;

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

Cf. A001263, A157114, A157117.

A001263:= func< n,k | Binomial(n-1,k-1)*Binomial(n,k)/(n-k+1) >;
f:= func< n,k | k le n select A001263(n*k+1,n-k+1) else A001263(n*(n-k)+1, k+1) >;
A157118:= func< n,k | n eq 1 select 1 else f(n,k) + f(n,n-k) >;
[A157118(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022

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

def A001263(n,k): return binomial(n-1,k-1)*binomial(n,k)/(n-k+1)
def f(n,k): return A001263(n*k+1,n-k+1) if (kA001263(n*(n-k)+1, k+1)
def A157118(n,k): return 1 if (n==1) else f(n,k) + f(n,n-k)
flatten([[A157118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022

T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1.

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

Edited by G. C. Greubel, Jan 11 2022

A178578 Diagonal sums of second binomial transform of the Narayana triangle A001263.

Original entry on oeis.org

1, 3, 10, 34, 118, 417, 1497, 5448, 20063, 74649, 280252, 1060439, 4040413, 15488981, 59701236, 231236830, 899559100, 3513314664, 13770811198, 54152480421, 213585706927, 844723104691, 3349274471386, 13310603555085, 53012829376985, 211560158583657, 845856494229348, 3387782725245302, 13590698721293800, 54604853170818121, 219706932640295523

Offset: 0

Paul Barry, Dec 26 2010

nonn

Hankel transform is the (1,-1) Somos-4 sequence A178079.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A025254.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 -3*x-x^2 - Sqrt(x^4+2*x^3+7*x^2-6*x+1))/(2*x^3))); // G. C. Greubel, Aug 14 2018

Table[Sum[Sum[Binomial[n-k,j]*Binomial[j,k]*Binomial[j+1,k]*2^(n-k-j)/(k+1),{j,0,n-k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Mar 02 2014 *)
CoefficientList[Series[(1-3*x-x^2 -Sqrt[x^4+2*x^3+7*x^2-6*x+1])/(2*x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)

a(n)=sum(k=0,floor(n/2), sum(j=0,n-k,binomial(n-k,j)*binomial(j,k)*binomial(j+1,k)*2^(n-k-j)/(k+1)));
vector(22,n,a(n-1))

a(n) = A025254(n+2).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k,j)*C(j,k)*C(j+1,k)*2^(n-k-j)/(k+1).
From Vaclav Kotesovec, Mar 02 2014: (Start)
Recurrence: (n+3)*a(n) = 3*(2*n+3)*a(n-1) - 7*n*a(n-2) - (2*n-3)*a(n-3) - (n-3)*a(n-4).
G.f.: (1 - 3*x - x^2 - sqrt(x^4 + 2*x^3 + 7*x^2 - 6*x + 1))/(2*x^3).
a(n) ~ (130-216*r-64*r^2-29*r^3) * sqrt(2*r^3+14*r^2-18*r+4) / (4 * sqrt(Pi) * n^(3/2) * r^n), where r = 1/6*(-3 + sqrt(3*(-11 + (1009 - 24*sqrt(183))^(1/3) + (1009 + 24*sqrt(183))^(1/3))) - sqrt(-66 - 3*(1009 - 24*sqrt(183))^(1/3) - 3*(1009 + 24*sqrt(183))^(1/3) + 216*sqrt(3/(-11 + (1009 - 24*sqrt(183))^(1/3) + (1009 + 24*sqrt(183))^(1/3))))) = 0.23742047190096998... is the root of the equation r^4 + 2*r^3 + 7*r^2 - 6*r + 1 = 0.
(End)

A376001 Numbers that can be written as a Narayana number (A001263) in at least 3 ways.

Original entry on oeis.org

1, 105, 1176, 4950, 5713890

Offset: 1

Pontus von Brömssen, Sep 06 2024

The first 5 terms are triangular numbers.
a(2), ..., a(5) can all be written as a Narayana number in exactly 4 ways.
a(6) > 2*10^35 (if it exists).

			With T(n,k) = A001263(n,k):
      105 = T( 7,3) = T( 7, 5) = T(  15,2) = T(  15,  14);
     1176 = T( 9,4) = T( 9, 6) = T(  49,2) = T(  49,  48);
     4950 = T(11,4) = T(11, 8) = T( 100,2) = T( 100,  99);
  5713890 = T(92,3) = T(92,90) = T(3381,2) = T(3381,3380).

Cf. A000217, A001263, A003015, A374796, A375573, A375999, A376000.

from math import isqrt
from bisect import insort
from itertools import islice
def A010054(n):
    return isqrt(m:=8*n+1)**2 == m
def A376001_generator():
    yield 1
    nkN_list = [(5, 3, 20)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        c = 0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0:
                if c >= 3 or A010054(N0): yield N0
                break
            central = n==2*k-1
            c += 2-central
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if central:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A376001_list(nmax):
    return list(islice(A376001_generator(),nmax))

A095801 Square of Narayana triangle A001263: View A001263 as a lower triangular matrix. Then the square of that matrix is also lower triangular. Sequence gives this lower triangle, read by rows.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 14, 30, 12, 1, 42, 140, 100, 20, 1, 132, 630, 700, 250, 30, 1, 429, 2772, 4410, 2450, 525, 42, 1, 1430, 12012, 25872, 20580, 6860, 980, 56, 1, 4862, 51480, 144144, 155232, 74088, 16464, 1680, 72, 1, 16796, 218790, 772200, 1081080, 698544

Offset: 1

Gary W. Adamson, Jun 07 2004

The first three columns are A000108 (the Catalan numbers), A002457 and A085374.

			The first 3 rows are 1; 2, 1; 5, 6, 1; since the first 3 rows of the Narayana triangle in matrix format are M = [1 0 0 / 1 1 0 / 1 3 1]. Then M^2 = [1 0 0 / 2 1 0 / 5 6 1].
Triangle starts:
   1;
   2,   1;
   5,   6,   1;
  14,  30,  12,  1;
  42, 140, 100, 20, 1;
  ...

Cf. A000108, A001263, A002457, A085374.

t[n_, k_] = Sum[1/(i*k)*(Binomial[i-1, k-1]*Binomial[i, k-1]* Binomial[n-1, i-1]*Binomial[n, i-1]), {i, k, n}];
Flatten[Table[t[n, k], {n, 1, 10}, {k, 1, n}]][[1;;50]] (* Jean-François Alcover, Jul 21 2011 *)

T(n, k) = Sum_{i = k..n} A001263(n, i)*A001263(i, k).

T(n, n-1) = n*(n-1).

Edited and extended by David Wasserman, Sep 24 2004

A105557 Row sums of triangle A105556, in which column n equals the row sums of A001263^n, which is the n-th matrix power of the Narayana triangle A001263.

Original entry on oeis.org

1, 2, 4, 10, 32, 128, 626, 3681, 25574, 206402, 1908996, 20024149, 236142157, 3106393358, 45265833590, 726249472784, 12761749378320, 244453274012442, 5082582988294164, 114258645210526486, 2767462674168199303

Offset: 0

Paul D. Hanna, Apr 14 2005

nonn

Cf. A001263, A105556.

{a(n)=local(N=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1,j-1)*binomial(m,j-1)/j))); sum(k=0,n,sum(j=0,n-k,(N^k)[n-k+1,j+1]))}

A105558 Central terms in even-indexed rows of triangle A105556 and thus equals the n-th row sum of the n-th matrix power of the Narayana triangle A001263.

Original entry on oeis.org

1, 2, 12, 148, 3105, 99156, 4481449, 272312216, 21414443481, 2116193061340, 256712977920256, 37506637787774112, 6496315164318118165, 1316230822119433518312, 308426950979497974254310

Offset: 0

Paul D. Hanna, Apr 14 2005

nonn

Each term a(n) is divisible by (n+1) for all n>=0.

			Terms a(n) divided by (n+1) begin:
1,1,4,37,621,16526,640207,34039027,2379382609,211619306134,...
Contribution from _Paul D. Hanna_, Jan 31 2009: (Start)
G.f.: A(x) = 1 + 2*x + 12*x^2/3 + 148*x^3/18 + 3105*x^4/180 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and:
F(x) = 1 + x + 4*x^2/3 + 37*x^3/18 + 621*x^4/180 + 16526*x^5/2700 +...+ A155926(n)*x^n/[n!*(n+1)!/2^n] +...
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 +...+ x^n/[n!*(n+1)!/2^n] +... (End)

Cf. A001263, A105556, A155926.

a(n)=local(N=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1,j-1)*binomial(m,j-1)/j))); sum(j=0,n,(N^n)[n+1,j+1])
for(n=0,20,print1(a(n),", "))

a(n)=local(F=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(deriv(serreverse(x/F)),n)*n!*(n+1)!/2^n
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Jan 31 2009

Contribution from Paul D. Hanna, Jan 31 2009: (Start)
a(n) = (n+1)*A155926(n) for n>=0.
G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and F(x) = Sum_{n>=0} A155926(n)*x^n/[n!*(n+1)!/2^n] with B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n] and A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n]. (End)

A112338 Triangle read by rows, generated from A001263.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 12, 14, 1, 1, 5, 22, 57, 42, 1, 1, 6, 35, 148, 303, 132, 1, 1, 7, 51, 305, 1144, 1743, 429, 1, 8, 70, 546, 3105, 9784, 10629, 1430, 1

Offset: 0

Gary W. Adamson, Sep 04 2005

Rows of the array are row sums of n-th powers of the Narayana triangle; e.g., row 1 = A000108: (1, 2, 5, 14, 42, ...); row 2 = row sums of the Narayana triangle squared (A103370): (1, 3, 12, 57, 303, ...), etc.

			In the array, antidiagonal terms (1, 3, 5, 1) become row 3 of the triangle.
First few rows of the array:
  1, 1,  1,   1,    1,     1, ...
  1, 2,  5,  14,   42,   132, ...
  1, 3, 12,  57,  303,  1743, ...
  1, 4, 22, 148, 1144,  9784, ...
  1, 5, 35, 305, 3105, 35505, ...
First few rows of the triangle:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  5,   1;
  1, 4, 12,  14,   1;
  1, 5, 22,  57,  42,   1;
  1, 6, 35, 148, 303, 132, 1;

Cf. A001263, A000326, A005915, A095266, A000108, A103370.

Let M be the infinite lower triangular Narayana triangle (A001263). Perform M^n * [1 0 0 0 ...] getting an array. Take antidiagonals of the array which become rows of the triangle A112338.

cp's OEIS Frontend

A132789 Triangle read by rows: T(n,k) = A007318(n-1, k-1) + A001263(n, k) - 1.

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A136536 Triangle read by rows: A001263 * A128064 * A000012 as infinite lower triangular matrices.

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A137940 Triangle read by rows, antidiagonals of an array formed by A000012 * A001263 (transform).

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A157118 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1, read by rows.

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A178578 Diagonal sums of second binomial transform of the Narayana triangle A001263.

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A376001 Numbers that can be written as a Narayana number (A001263) in at least 3 ways.

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A095801 Square of Narayana triangle A001263: View A001263 as a lower triangular matrix. Then the square of that matrix is also lower triangular. Sequence gives this lower triangle, read by rows.

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A105557 Row sums of triangle A105556, in which column n equals the row sums of A001263^n, which is the n-th matrix power of the Narayana triangle A001263.

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A157118 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(nk+1, n-k+1) if k <= n otherwise A001263(n(n-k)+1, k+1) and T(1, k) = 1, read by rows.