A033282 -id:A033282 - cp's OEIS frontend

A243661 Triangle read by rows: the x = 1+q Narayana triangle at m=3.

1, 4, 3, 22, 33, 12, 140, 315, 231, 55, 969, 2907, 3213, 1547, 273, 7084, 26565, 39270, 28560, 10200, 1428, 53820, 242190, 448500, 437000, 235980, 66861, 7752, 420732, 2208843, 4916457, 6009003, 4351347, 1864863, 437437, 43263, 3362260, 20173560, 52451256, 77134200, 70122000, 40320150, 14307150, 2861430, 246675

Offset: 1

N. J. A. Sloane, Jun 13 2014

See Novelli-Thibon (2014) for precise definition.

			Triangle begins:
     1;
     4,     3;
    22,    33,    12;
   140,   315,   231,    55;
   969,  2907,  3213,  1547,   273;
  7084, 26565, 39270, 28560, 10200,  1428;
  ...

Lars Blomberg, Table of n, a(n) for n = 1..210 (The first 20 rows)
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 9.

The left column is A002293, the main diagonal is A001764.
The case m=1 is A126216 or A033282 (its mirror image).
The case m=2 is A243660.

polrecip[P_, x_] := P /. x -> 1/x // Together // Numerator;
P[n_, m_] := Sum[Binomial[m n + 1, k] Binomial[(m + 1) n - k, n - k] (1 - x)^k x^(n - k), {k, 0, n}]/(m n + 1);
T[m_] := Reap[For[i=1, i <= 20, i++, z = polrecip[P[i, m], x] /. x -> 1+q; Sow[CoefficientList[z, q]]]][[2, 1]];
T[3] // Flatten (* Jean-François Alcover, Oct 08 2018, from PARI *)

N(n,m)=sum(k=0,n,binomial(m*n+1,k)*binomial((m+1)*n-k,n-k)*(1-x)^k*x^(n-k))/(m*n+1);
T(m)=for(i=1,20,z=subst(polrecip(N(i,m)),x,1+q);print(Vecrev(z)));
T(3) /* Lars Blomberg, Jul 17 2017 */

From Werner Schulte, Nov 23 2018: (Start)
T(n,k) = binomial(4*n+1-k,n-k) * binomial(3*n,k-1) / n.
More generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(m*n,k-1) / n where m = 3.
Sum_{k=1..n} (-1)^k * T(n,k) = -1. (End)
Sum_{k = 1..n} (-1)^(k+1)*T(n,k)*binomial(x + 4*n - k + 1, 4*n - k + 1) = (x + 1) * ( Product_{k = 2..n} (x + k)^2 ) * ( Product_{k = 1..2*n+1} (x + n + k) ) / (n!*(3*n + 1)!) for n >= 1. Cf. A126216 and A243660. - Peter Bala, Oct 08 2022

a(22)-a(39) from Lars Blomberg, Jul 12 2017

A079508 Triangle T(n,k) (n >= 2, k >= 1) of Raney numbers read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 5, 1, 0, 0, 5, 9, 1, 0, 0, 0, 21, 14, 1, 0, 0, 0, 14, 56, 20, 1, 0, 0, 0, 0, 84, 120, 27, 1, 0, 0, 0, 0, 42, 300, 225, 35, 1, 0, 0, 0, 0, 0, 330, 825, 385, 44, 1, 0, 0, 0, 0, 0, 132, 1485, 1925, 616, 54, 1, 0, 0, 0, 0, 0, 0, 1287, 5005, 4004, 936, 65, 1

Offset: 2

N. J. A. Sloane, Jan 21 2003

There are only m nonzero entries in the m-th column.

Related to A033282: shift row n of A033282 triangle n places to the right and transpose the resulting table. - Michel Marcus, Feb 04 2014

			From _Michel Marcus_, Feb 04 2014: (Start)
Triangle starts:
  1;
  0, 1;
  0, 2, 1;
  0, 0, 5,  1;
  0, 0, 5,  9,  1;
  0, 0, 0, 21, 14,   1;
  0, 0, 0, 14, 56,  20,    1;
  0, 0, 0,  0, 84, 120,   27,    1;
  0, 0, 0,  0, 42, 300,  225,   35,   1;
  0, 0, 0,  0,  0, 330,  825,  385,  44,  1;
  0, 0, 0,  0,  0, 132, 1485, 1925, 616, 54, 1;
  ... (End)

G. C. Greubel, Rows n=2..100 of triangle, flattened
Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30 (see definition p. 26 and table p. 27).
G. N. Raney, Functional composition patterns and power series reversion, Trans. Amer. Math. Soc., 94 (1960), pp. 441-451.

Row sums give A005043.
Column sums give A001003.
Alternating sum of each column is 1.
Second diagonal on right gives A000096.
Central terms give A000108.
Cf. A033282, A126216 (transposed variants).

Flat(List([1..10], n->List([1..n-1], k-> Binomial(k,n-k)*Binomial(n ,k+1)/k ))); # G. C. Greubel, Jan 17 2019

[[Binomial(k,n-k)*Binomial(n,k+1)/k: k in [1..n-1]]: n in [2..10]]; // G. C. Greubel, Jan 17 2019

Table[Binomial[k, n-k]*Binomial[n, k+1]/k, {n,2,10}, {k,1,n-1}]//Flatten (* G. C. Greubel, Jan 17 2019 *)

tabl(nn) = {for (n = 2, nn, for (k = 1, n-1, print1(binomial(k, n-k)*binomial(n, k+1)/k, ", ");); print(););} \\ Michel Marcus, Feb 04 2014

[[binomial(k,n-k)*binomial(n,k+1)/k for k in (1..n-1)] for n in (2..10)] # G. C. Greubel, Jan 17 2019

T(n,k) = binomial(k, n-k) * binomial(n, k+1)/k. - Michel Marcus, Feb 04 2014
From Andrew Howroyd, Jan 24 2025: (Start)
T(n,k) = A033282(k + 2, n - k - 1) = A126216(k, 2*k - n).
G.f.: -1 + ((1 + y*x) - sqrt(1 - 2*y*x + (y^2 - 4*y)*x^2))/(2*x*y*(1 + x)).
G.f.: -1 + (1/(x*y))*Series_Reversion(x*(1 - x)/(y - y*x + x^2)). (End)

Corrected and extended by Michel Marcus, Feb 04 2014

A183759 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a circle, triangle decomposed by number of chords.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 2, 7, 7, 1, 3, 15, 33, 22, 1, 3, 24, 94, 136, 66, 1, 4, 41, 218, 525, 552, 217, 1, 4, 57, 440, 1517, 2673, 2233, 715, 1, 5, 86, 804, 3748, 9439, 12917, 8807, 2438

Offset: 1

David Scambler, Jan 06 2011

Decomposition of A183757 by number of chords (pairs of parentheses). Row sums are A183757. See A183757 for description.
Triangle is
1
1 1
1 2  3
1 2  7  7
1 3 15 33  22
1 3 24 94 136 66
...
Right diagonal is apparently A007595.

Decomposition of A183757.

Cf. analogous decomposition of super-Catalan numbers in triangle A033282.

A183760 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a flippable circle, triangle decomposed by number of chords.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 5, 4, 1, 3, 10, 18, 12, 1, 3, 16, 51, 71, 34, 1, 4, 26, 116, 271, 280, 111, 1, 4, 36, 233, 776, 1349, 1124, 360, 1, 5, 53, 421, 1908, 4749, 6487, 4415, 1226

Offset: 1

David Scambler, Jan 06 2011

Decomposition of A183758 by number of chords (pairs of parentheses). Row sums are A183758. See A183758 for description.
Triangle is
1
1 1
1 2  2
1 2  5  4
1 3 10 18 12
1 3 16 51 71 34
...
Right diagonal is apparently A001895.

Decomposition of A183758. Cf. analogous decomposition of super-Catalan numbers in triangle A033282.

A357613 Triangle read by rows T(n, k) = binomial(2n, k) binomial(3n - k, 2n).

Original entry on oeis.org

1, 3, 2, 15, 20, 6, 84, 168, 105, 20, 495, 1320, 1260, 504, 70, 3003, 10010, 12870, 7920, 2310, 252, 18564, 74256, 120120, 100100, 45045, 10296, 924, 116280, 542640, 1058148, 1113840, 680680, 240240, 45045, 3432

Offset: 0

F. Chapoton, Oct 06 2022

Each line should be the f-vector of a cellular complex. The sequence seems to give the coefficients in a binomial basis of the integer-valued polynomials (x+1)*(x+2)*...*(x+2*n)*(x+1)*(x+2)*...*(x+n)/(n!*(2n)!).

The precise expansion is (x+1)*(x+2)*...*(x+2*n)*(x+1)*(x+2)*...*(x+n)/(n!*(2*n)!) = Sum_{k = 0..n} (-1)^k*T(n,k)*binomial(x+3*n-k, 3*n-k), as can be verified using the WZ algorithm. For example, n = 3 gives (x+1)^2*(x+2)^2*(x+3)^2*(x+4)*(x+5)*(x+6)/(3!*6!) = 84*binomial(x+9, 9) - 168*binomial(x+8, 8) + 105*binomial(x+7, 7) - 20*binomial(x+6, 6). - Peter Bala, Jun 25 2023

			As a triangle of numbers, this starts with
  1;
  3, 2;
  15, 20, 6;
  84, 168, 105, 20;
  495, 1320, 1260, 504, 70.
Here is an example for n=1 as coefficients (up to sign) in the binomial basis of integer-valued polynomials:
(x+1)*(x+2)*(x+1)/2 = 3*binomial(x+3,3)-2*binomial(x+2,2).

Row sums A026000. Cf. A000984, A005809 (k=0), A144485 (k=1), A033282, A110608, A243660.

A357613 := proc(n,k)
    binomial(2*n,k)*binomial(3*n-k,2*n) ;
end proc:
seq(seq(A357613(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jul 06 2023

Table[Binomial[2n,k]Binomial[3n-k,2n],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Oct 11 2023 *)

def a(n):
    return [binomial(2 * n, k) * binomial(3 * n - k, 2 * n)
            for k in range(n + 1)]

T(n,k) = binomial(2*n, k) * binomial(3*n - k, 2*n) for 0 <= k <= n

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A243661 Triangle read by rows: the x = 1+q Narayana triangle at m=3.

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A079508 Triangle T(n,k) (n >= 2, k >= 1) of Raney numbers read by rows.

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A183759 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a circle, triangle decomposed by number of chords.

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A183760 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a flippable circle, triangle decomposed by number of chords.

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A357613 Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).

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A357613 Triangle read by rows T(n, k) = binomial(2n, k) binomial(3n - k, 2n).