A126216 -id:A126216 - cp's OEIS frontend

A243660 Triangle read by rows: the x = 1+q Narayana triangle at m=2.

1, 3, 2, 12, 16, 5, 55, 110, 70, 14, 273, 728, 702, 288, 42, 1428, 4760, 6160, 3850, 1155, 132, 7752, 31008, 50388, 42432, 19448, 4576, 429, 43263, 201894, 395010, 418950, 259350, 93366, 18018, 1430, 246675, 1315600, 3010700, 3853696, 3010700, 1466080, 433160, 70720, 4862

Offset: 1

N. J. A. Sloane, Jun 13 2014

See Novelli-Thibon (2014) for precise definition.
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*...*(x+2n+1) / (n! * (2n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 09 2022
The Maple code T(n,k) := binomial(3*n+1-k,n-k)*binomial(2*n,k-1)/n: with(sumtools): sumrecursion( (-1)^(k+1)*T(n,k)*binomial(x+3*n-k+1, 3*n-k+1), k, s(n) ); returns the recurrence 2*(2*n+1)*n^2*s(n) = (x+n)*(x+2*n)*(x+2*n+1)*s(n-1). The above observation follows from this. - Peter Bala, Oct 30 2022

			Triangle begins:
     1;
     3,    2;
    12,   16,    5;
    55,  110,   70,   14;
   273,  728,  702,  288,   42;
  1428, 4760, 6160, 3850, 1155,  132;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
J.-C. Novelli and 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. 8.

Row sums give A034015(n-1).
The case m=1 is A126216 or A033282 (its mirror image).
The case m=3 is A243661.
The right diagonal is A000108.
The left column is A001764.
Same table as A383453, transposed. - Dan Eilers, May 06 2025

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[2] // 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(2)  /* Lars Blomberg, Jul 17 2017 */

T(n,k) = binomial(3*n+1-k,n-k) * binomial(2*n,k-1) / n; \\ Andrew Howroyd, Nov 23 2018

From Werner Schulte, Nov 23 2018: (Start)
T(n,k) = binomial(3*n+1-k,n-k) * binomial(2*n,k-1) / n.
More generally: T(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(m*n,k-1) / n, where m = 2.
Sum_{k=1..n} (-1)^k * T(n,k) = -1. (End)

Corrected example and a(22)-a(43) from Lars Blomberg, Jul 12 2017

a(44)-a(45) from Werner Schulte, Nov 23 2018

A126182 Let P be Pascal's triangle A007318 and let N be Narayana's triangle A001263, both regarded as lower triangular matrices. Sequence gives triangle obtained from P*N, read by rows.

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 18, 9, 1, 16, 56, 50, 14, 1, 32, 160, 220, 110, 20, 1, 64, 432, 840, 645, 210, 27, 1, 128, 1120, 2912, 3150, 1575, 364, 35, 1, 256, 2816, 9408, 13552, 9534, 3388, 588, 44, 1, 512, 6912, 28800, 53088, 49644, 24822, 6636, 900, 54, 1

Offset: 0

Emeric Deutsch, Dec 19 2006, Mar 30 2007

Also T(n,k) is number of hex trees with n edges and k left edges (0<=k<=n). A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a median child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference). Accordingly, one can have left, vertical, or right edges.
Also (with a different offset) T(n,k) is the number of skew Dyck paths of semilength n and having k peaks (1<=k<=n). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. E.g., T(3,2)=5 because we have (UD)U(UD)D, (UD)U(UD)L, U(UD)D(UD), U(UD)(UD)D and U(UD)(UD)L (the peaks are shown between parentheses).
Sum of terms in row n = A002212(n+1). T(n,1) = A001793(n); T(n,2) = A006974(n-2); Sum_{k=0..n}kT(n,k) = A026379(n+1).
A126216 = N * P. - Gary W. Adamson, Nov 30 2007

			The triangle P begins
  1,
  1, 1
  1, 2, 1
  1, 3, 3, 1, ...
and T begins
  1,
  1,  1,
  1,  3,  1,
  1,  6,  6,  1,
  1, 10, 20, 10, 1, ...
The product P*T gives
   1,
   2,  1,
   4,  5,  1,
   8, 18,  9,  1,
  16, 56, 50, 14, 1, ...

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Cf. A001263, A007318, A002212, A001793, A006974, A026379.

Cf. A128718, A026378, A126216.

T:=proc(n,k) if k=0 then 2^n elif k<=n then binomial(n+1,k)*sum(binomial(k,n-k-j)*binomial(n+1-k,j)*2^j,j=n-2*k..n-k)/(n+1) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

t[n_, 0] := 2^n; t[n_, k_] := Binomial[n+1, k]*Sum[Binomial[k, n-k-j]*Binomial[n+1-k, j]*2^j, {j, n-2*k, n-k}]/(n+1); Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 12 2013 *)
nmax = 10; n[x_, y_] := (1-x*(1+y) - Sqrt[(1-x*(1+y))^2 - 4*y*x^2])/(2*x); s = Series[(n[x/(1-x), y]-1)/x, {x, 0, nmax+1}, {y, 0, nmax+1}];t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, nmax}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Apr 16 2015, after Vladimir Kruchinin *)

tabl(nn) = {mP = matrix(nn, nn, n, k, binomial(n-1, k-1)); mN = matrix(nn, nn, n, k, binomial(n-1, k-1) * binomial(n, k-1) / k); mprod = mP*mN; for (n=1, nn, for (k=1, n, print1(mprod[n, k], ", ");); print(););} \\ Michel Marcus, Apr 16 2015

T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=n-2k..n-k}2^j*binomial(k,n-k-j)*binomial(n+1-k,j) if 0 < k <= n; T(n,0) = 2^n.
G.f. G=G(t,z) satisfies G  = 1 + (t+2)*z*G + t*z^2*G^2.
E.g.f.: exp((t+2)*x)*BesselI_{1}(2*sqrt(t)*x)/(sqrt(t)*x). - Peter Luschny, Oct 29 2014
G.f.: N(x/(1-x),y)-1)/x, where N(x,y) is the g.f. of Narayana's triangle A001263. - Vladimir Kruchinin, Apr 06 2015.

New definition in terms of P and N from Philippe Deléham, Jun 30 2007

Edited by N. J. A. Sloane, Jul 22 2007

A269732 Dimensions of the 4-polytridendriform operad TDendr_4.

Original entry on oeis.org

1, 9, 101, 1269, 17081, 240849, 3511741, 52515549, 801029681, 12414177369, 194922521301, 3094216933509, 49575333021801, 800645021406369, 13020241953611181, 213025792632813549, 3504075376813414241, 57914491106005287849, 961297812844696640581, 16017765308027639317269, 267831397282643166904601, 4492625888792276208945009, 75578709400747348254905501

Offset: 1

N. J. A. Sloane, Mar 08 2016

Gheorghe Coserea, Table of n, a(n) for n = 1..512
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016; Adv. Appl. Math., 77, 3-85, 2016.

Cf. A001003, A005060, A001263, A126216, A269730, A269731, A084769.

I:=[1,9]; [n le 2 select I[n] else (9*(2*n-1)*Self(n-1)-(n-2)*Self(n-2))/(n+1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2016

Rest[CoefficientList[Series[(1 - 9*x - Sqrt[1 - 18*x + x^2])/(40*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 9]/(2*Sqrt[5]), {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 9, (n+1) a[n] == 9 (2 n - 1) a[n-1] - (n - 2) a[n-2]}, a, {n, 25}] (* Vincenzo Librandi, Nov 29 2016 *)

A001263(n,k) = binomial(n-1,k-1) * binomial(n, k-1)/k;
dimTDendr(n,q) = sum(k = 0, n-1, (q+1)^k * q^(n-k-1) * A001263(n,k+1));
my(q=4); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016

my(q=4, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

a(n) = P_n(4), where P_n(x) is the polynomial associated with row n of triangle A126216 in order of decreasing powers of x.
Recurrence: (n+1)*a(n) = 9*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(40 + 18*sqrt(5)) * (9 + 4*sqrt(5))^n / (40*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ phi^(6*n + 3) / (2^(5/2) * 5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 23 2017
A(x) = -serreverse(A005060(x))(-x). - Gheorghe Coserea, Sep 30 2017
O.g.f.: A(x) = (1 - sqrt(1 - 18*x + x^2) - 9*x)/(40*x). - Peter Bala, Jan 25 2018
From Peter Bala, Dec 25 2020: (Start)
a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 4.
a(n) = (1/(2*n+1)) * (1/(2*m*(m+1))) * ( Legendre_P(n+1,2*m+1) - Legendre_P(n-1,2*m+1) ) at m = 4. (End)

More terms from Gheorghe Coserea, Apr 23 2016

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

Original entry on oeis.org

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

A269730 Dimensions of the 2-polytridendriform operad TDendr_2.

Original entry on oeis.org

1, 5, 31, 215, 1597, 12425, 99955, 824675, 6939769, 59334605, 513972967, 4501041935, 39784038517, 354455513105, 3179928556219, 28701561707675, 260447708523505, 2374690737067925, 21744508765633327, 199877846477679815, 1843718766426242221, 17060955558786455705, 158333204443000060291

Offset: 1

N. J. A. Sloane, Mar 08 2016

Gheorghe Coserea, Table of n, a(n) for n = 1..512
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016; Adv. Appl. Math., 77, 3-85, 2016.

Cf. A001047, A001263, A126216, A269731, A269732, A001003, A006442.

Rest[CoefficientList[Series[(1 - 5*x - Sqrt[1 - 10*x + x^2])/(12*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 5]/Sqrt[6], {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)

A001263(n,k) = binomial(n-1,k-1) * binomial(n, k-1)/k;
dimTDendr(n,q) = sum(k = 0, n-1, (q+1)^k * q^(n-k-1) * A001263(n,k+1));
my(q=2); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016

my(q=2, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

a(n) = P_n(2), where P_n(x) is the polynomial associated with row n of triangle A126216 in order of decreasing powers of x.
Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(12 + 5*sqrt(6)) * (5 + 2*sqrt(6))^n / (12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
A(x) = -serreverse(A001047(x))(-x). - Gheorghe Coserea, Sep 30 2017
From Peter Bala, Dec 25 2020: (Start)
a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 2.
a(n) = (1/(2*n+1)) * (1/(2*m*(m+1))) * ( Legendre_P(n+1,2*m+1) - Legendre_P(n-1,2*m+1) ) at m = 2. (End)
G.f. A(x) = x*exp( Sum_{n >= 1} A006442(n)*x^n/n ). - Peter Bala, Jan 09 2022

More terms from Gheorghe Coserea, Apr 23 2016

A269731 Dimensions of the 3-polytridendriform operad TDendr_3.

Original entry on oeis.org

1, 7, 61, 595, 6217, 68047, 770149, 8939707, 105843409, 1273241431, 15517824973, 191202877411, 2377843390873, 29807864423071, 376255282112629, 4778240359795147, 61007205215610529, 782648075371992487, 10083436451634033757, 130413832663780730995, 1692599303723819234281, 22037570163808433691247, 287762084009227350367621

Offset: 1

N. J. A. Sloane, Mar 08 2016

Gheorghe Coserea, Table of n, a(n) for n = 1..512
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016; Adv. Appl. Math., 77, 3-85, 2016.

Cf. A001003, A005061, A001263, A126216, A269730, A269732.

I:=[1,7]; [n le 2 select I[n] else (7*(2*n-1)*Self(n-1)-(n-2)*Self(n-2))/(n+1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2016

Rest[CoefficientList[Series[(1 - 7*x - Sqrt[1 - 14*x + x^2])/(24*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 7]/(2*Sqrt[3]), {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 7, (n + 1) a[n] == 7 (2 n - 1) a[n-1] - (n - 2) a[n-2]}, a, {n, 25}] (* Vincenzo Librandi, Nov 29 2016 *)

A001263(n,k) = binomial(n-1,k-1) * binomial(n, k-1)/k;
dimTDendr(n,q) = sum(k = 0, n-1, (q+1)^k * q^(n-k-1) * A001263(n,k+1));
my(q=3); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016

my(q=3, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

a(n) = P_n(3), where P_n(x) is the polynomial associated with row n of triangle A126216 in order of decreasing powers of x.
Recurrence: (n+1)*a(n) = 7*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(24 + 14*sqrt(3)) * (7 + 4*sqrt(3))^n / (24*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
A(x) = -serreverse(A005061(x))(-x). - Gheorghe Coserea, Sep 30 2017
From Peter Bala, Dec 25 2020: (Start)
a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 3.
a(n) = (1/(2*n+1)) * (1/(2*m*(m+1))) * ( Legendre_P(n+1,2*m+1) - Legendre_P(n-1,2*m+1) ) at m = 3. (End)

More terms from Gheorghe Coserea, Apr 23 2016

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

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A243660 Triangle read by rows: the x = 1+q Narayana triangle at m=2.

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A126182 Let P be Pascal's triangle A007318 and let N be Narayana's triangle A001263, both regarded as lower triangular matrices. Sequence gives triangle obtained from P*N, read by rows.

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A269732 Dimensions of the 4-polytridendriform operad TDendr_4.

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

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A269730 Dimensions of the 2-polytridendriform operad TDendr_2.

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A269731 Dimensions of the 3-polytridendriform operad TDendr_3.

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