A103210 -id:A103210 - cp's OEIS frontend

A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0, 1, 55, 825, 4950, 13860

Offset: 0

Philippe Deléham, Jan 19 2004

Number of Dyck n-paths with exactly k peaks. - Peter Luschny, May 10 2014

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 1,  3,   1;
[4] 0, 1,  6,   6,    1;
[5] 0, 1, 10,  20,   10,    1;
[6] 0, 1, 15,  50,   50,   15,    1;
[7] 0, 1, 21, 105,  175,  105,   21,   1;
[8] 0, 1, 28, 196,  490,  490,  196,  28,  1;
[9] 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1;

Indranil Ghosh, Rows 0..100, flattened
Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
FindStat - Combinatorial Statistic Finder, The number of peaks of a Dyck path, The number of double rises of a Dyck path, The number of valleys of a Dyck path, The number of left oriented leafs except the first one of a binary tree, The number of left tunnels of a Dyck path
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Mirror image of triangle A131198. A000108 (row sums, Catalan).
Columns: A000217, A002415, A006542, A006857.
Cf. A001263, A084938, A243752.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n) for x=0,1,2,3,4,5,6,7,8,9. - Philippe Deléham, Aug 10 2006
Sum_{k=0..n} x^(n-k)*T(n,k) = A090192(n+1), A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Oct 21 2006
Sum_{k=0..n} T(n,k)*x^k*(x-1)^(n-k) = A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Oct 20 2007

[[(&+[(-1)^(j-k)*Binomial(2*n-j,j)*Binomial(j,k)*Binomial(2*n-2*j,n-j)/(n-j+1): j in [0..n]]): k in [0..n]]: n in [0..10]];

A090181 := (n,k) -> binomial(n,n-k)*binomial(n-1,n-k)/(n-k+1):
seq(print( seq(A090181(n,k),k=0..n)),n=0..5); # Peter Luschny, May 10 2014
egf := 1+int((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x,x);
s := n -> n!*coeff(series(egf,x,n+2),x,n);
seq(print(seq(coeff(s(n),t,j),j=0..n)),n=0..9); # Peter Luschny, Oct 30 2014
T := proc(n, k) option remember; if k = n or k = 1 then 1 elif k < 1 then 0 else (2*n/k - 1) * T(n-1, k-1) + T(n-1, k) fi end:
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Dec 31 2024

Flatten[Table[Sum[(-1)^(j-k) * Binomial[2n-j,j] * Binomial[j,k] * CatalanNumber[n-j], {j, 0, n}], {n,0,11},{k,0,n}]] (* Indranil Ghosh, Mar 05 2017 *)
p[0, ] := 1; p[1, x] := x; p[n_, x_] := ((2 n - 1) (1 + x) p[n - 1, x] - (n - 2) (x - 1)^2 p[n - 2, x]) / (n + 1);
Table[CoefficientList[p[n, x], x], {n, 0, 9}] // TableForm (* Peter Luschny, Apr 26 2022 *)

c(n) = binomial(2*n,n)/ (n+1);
tabl(nn) = {for(n=0, nn, for(k=0, n, print1(sum(j=0, n, (-1)^(j-k) * binomial(2*n-j,j) * binomial(j,k) * c(n-j)),", ");); print(););};
tabl(11); \\ Indranil Ghosh, Mar 05 2017

from functools import cache
@cache
def Trow(n):
    if n == 0: return [1]
    if n == 1: return [0, 1]
    if n == 2: return [0, 1, 1]
    A = Trow(n - 2) + [0, 0]
    B = Trow(n - 1) + [1]
    for k in range(n - 1, 1, -1):
        B[k] = (((B[k] + B[k - 1]) * (2 * n - 1)
               - (A[k] - 2 * A[k - 1] + A[k - 2]) * (n - 2)) // (n + 1))
    return B
for n in range(10): print(Trow(n)) # Peter Luschny, May 02 2022

def A090181_row(n):
    U = [0]*(n+1)
    for d in DyckWords(n):
        U[d.number_of_peaks()] +=1
    return U
for n in range(8): A090181_row(n) # Peter Luschny, May 10 2014

Triangle T(n, k), read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938. T(0, 0) = 1, T(n, 0) = 0 for n>0, T(n, k) = C(n-1, k-1)*C(n, k-1)/k for k>0.
Sum_{j>=0} T(n,j)*binomial(j,k) = A060693(n,k). - Philippe Deléham, May 04 2007
Sum_{k=0..n} T(n,k)*10^k = A143749(n+1). - Philippe Deléham, Oct 14 2008
From Paul Barry, Nov 10 2008: (Start)
Coefficient array of the polynomials P(n,x) = x^n*2F1(-n,-n+1;2;1/x).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*C(2n-j,j)*C(j,k)*A000108(n-j). (End)
Sum_{k=0..n} T(n,k)*5^k*3^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
Sum_{k=0..n} T(n,k)*(-2)^k = A152681(n); Sum_{k=0..n} T(n,k)*(-1)^k = A105523(n). - Philippe Deléham, Feb 03 2009
Sum_{k=0..n} T(n,k)*2^(n+k) = A156017(n). - Philippe Deléham, Nov 27 2011
T(n, k) = C(n,n-k)*C(n-1,n-k)/(n-k+1). - Peter Luschny, May 10 2014
E.g.f.: 1+Integral((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x dx). - Peter Luschny, Oct 30 2014
G.f.: (1+x-x*y-sqrt((1-x*(1+y))^2-4*y*x^2))/(2*x). - Alois P. Heinz, Nov 28 2021, edited by Ron L.J. van den Burg, Dec 19 2021
T(n, k) = [x^k] (((2*n - 1)*(1 + x)*p(n-1, x) - (n - 2)*(x - 1)^2*p(n-2, x))/(n + 1)) with p(0, x) = 1 and p(1, x) = x. - Peter Luschny, Apr 26 2022
Recursion based on rows (see the Python program):
T(n, k) = (((B(k) + B(k-1))*(2*n - 1) - (A(k) - 2*A(k-1) + A(k-2))*(n-2))/(n+1)), where A(k) = T(n-2, k) and B(k) = T(n-1, k), for n >= 3. # Peter Luschny, May 02 2022

A088617 Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 5, 1, 10, 30, 35, 14, 1, 15, 70, 140, 126, 42, 1, 21, 140, 420, 630, 462, 132, 1, 28, 252, 1050, 2310, 2772, 1716, 429, 1, 36, 420, 2310, 6930, 12012, 12012, 6435, 1430, 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862

Offset: 0

N. J. A. Sloane, Nov 23 2003

Row sums: A006318 (Schroeder numbers). Essentially same as triangle A060693 transposed.
T(n,k) is number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k U's. E.g., T(2,1)=3 because we have UHD, UDH and HUD. - Emeric Deutsch, Dec 06 2003
Little Schroeder numbers A001003 have a(n) = Sum_{k=0..n} A088617(n,k)*(-1)^(n-k)*2^k. - Paul Barry, May 24 2005
Conjecture: The expected number of U's in a Schroeder n-path is asymptotically Sqrt[1/2]*n for large n. - David Callan, Jul 25 2008
T(n, k) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha) = |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
The antidiagonals of this lower triangular matrix are the rows of A055151. - Tom Copeland, Jun 17 2015

			Triangle begins:
  [0] 1;
  [1] 1,  1;
  [2] 1,  3,   2;
  [3] 1,  6,  10,    5;
  [4] 1, 10,  30,   35,    14;
  [5] 1, 15,  70,  140,   126,    42;
  [6] 1, 21, 140,  420,   630,   462,   132;
  [7] 1, 28, 252, 1050,  2310,  2772,  1716,   429;
  [8] 1, 36, 420, 2310,  6930, 12012, 12012,  6435,  1430;
  [9] 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862;

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 16.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Hsien-Kuei Hwang and Satoshi Kuriki, Integrated empirical measures and generalizations of classical goodness-of-fit statistics, arXiv:2404.06040 [math.ST], 2024. See p. 11.
C. Jordan, Calculus of Finite Differences, Budapest, 1939. [Annotated scans of pages 448-450 only]
A. Kirillov, On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials, arXiv preprint arXiv:1502.00426 [math.RT], 2016.
M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.
Paul W. Lapey and Aaron Williams, A Shift Gray Code for Fixed-Content Łukasiewicz Words, Williams College, 2022.
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.
Jason P. Smith, The poset of graphs ordered by induced containment, arXiv:1806.01821 [math.CO], 2018.

Diagonals give A000217, A034827, A000910, A088625, A088626, A000108, A001700, A002457, A002802, A002803.
Cf. A000108, A001003, A006318, A060693, A084938, A085478.
Cf. A033282, A055151, A123160.
Cf. A001263, A011973, A055151, A060693, A074909, A086810, A107131.

[[Binomial(n+k,n)*Binomial(n,k)/(k+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 18 2015

R := n -> simplify(hypergeom([-n, n + 1], [2], -x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
seq(print(Trow(n)), n = 0..9); # Peter Luschny, Apr 26 2022

Table[Binomial[n+k, n] Binomial[n, k]/(k+1), {n,0,10}, {k,0,n}]//Flatten (* Michael De Vlieger, Aug 10 2017 *)

{T(n, k)= if(k+1, binomial(n+k, n)*binomial(n, k)/(k+1))}

flatten([[binomial(n+k, 2*k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 22 2022

Triangle T(n, k) read by rows; given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
T(n, k) = A085478(n, k)*A000108(k); A000108 = Catalan numbers. - Philippe Deléham, Dec 05 2003
Sum_{k=0..n} T(n, k)*x^k*(1-x)^(n-k) = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Aug 18 2005
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Oct 18 2007
O.g.f. (with initial 1 excluded) is the series reversion with respect to x of (1-t*x)*x/(1+x). Cf. A062991 and A089434. - Peter Bala, Jul 31 2012
G.f.: 1 + (1 - x - T(0))/y, where T(k) = 1 - x*(1+y)/( 1 - x*y/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013
From Peter Bala, Jul 20 2015: (Start)
O.g.f. A(x,t) = ( 1 - x - sqrt((1 - x)^2 - 4*x*t) )/(2*x*t) = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2 + ....
1 + x*(dA(x,t)/dx)/A(x,t) = 1 + (1 + t)*x + (1 + 4*t + 3*t^2)*x^2 + ... is the o.g.f. for A123160.
For n >= 1, the n-th row polynomial equals (1 + t)/(n+1)*Jacobi_P(n-1,1,1,2*t+1). Removing a factor of 1 + t from the row polynomials gives the row polynomials of A033282. (End)
From Tom Copeland, Jan 22 2016: (Start)
The o.g.f. G(x,t) = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/2x = (t + t^2) x + (t + 3t^2 + 2t^3) x^2 + (t + 6t^2 + 10t^3 + 5t^3) x^3 + ... generating shifted rows of this entry, excluding the first, was given in my 2008 formulas for A033282 with an o.g.f. f1(x,t) = G(x,t)/(1+t) for A033282. Simple transformations presented there of f1(x,t) are related to A060693 and A001263, the Narayana numbers. See also A086810.
The inverse of G(x,t) is essentially given in A033282 by x1, the inverse of f1(x,t): Ginv(x,t) = x [1/(t+x) - 1/(1+t+x)] = [((1+t) - t) / (t(1+t))] x - [((1+t)^2 - t^2) / (t(1+t))^2] x^2 + [((1+t)^3 - t^3) / (t(1+t))^3] x^3 - ... . The coefficients in t of Ginv(xt,t) are the o.g.f.s of the diagonals of the Pascal triangle A007318 with signed rows and an extra initial column of ones. The numerators give the row o.g.f.s of signed A074909.
Rows of A088617 are shifted columns of A107131, whose reversed rows are the Motzkin polynomials of A055151, related to A011973. The diagonals of A055151 give the rows of A088671, and the antidiagonals (top to bottom) of A088617 give the rows of A107131 and reversed rows of A055151. The diagonals of A107131 give the columns of A055151. The antidiagonals of A088617 (bottom to top) give the rows of A055151.
(End)
T(n, k) = [x^k] hypergeom([-n, 1 + n], [2], -x). - Peter Luschny, Apr 26 2022

A060693 Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 14, 35, 30, 10, 1, 42, 126, 140, 70, 15, 1, 132, 462, 630, 420, 140, 21, 1, 429, 1716, 2772, 2310, 1050, 252, 28, 1, 1430, 6435, 12012, 12012, 6930, 2310, 420, 36, 1, 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1, 16796

Offset: 0

F. Chapoton, Apr 20 2001

The rows sum to A006318 (Schroeder numbers), the left column is A000108 (Catalan numbers); the next-to-left column is A001700, the alternating sum in each row but the first is 0.
T(n,k) is the number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. E.g., T(n,k) = binomial(n,k)*binomial(2n-k,n-1)/n for n>0. - Emeric Deutsch, Dec 06 2003
A090181*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 14 2008
T(n,k) is also the number of rooted plane trees with maximal degree 3 and k vertices of degree 2 (a node may have at most 2 children, and there are exactly k nodes with 1 child). Equivalently, T(n,k) is the number of syntactically different expressions that can be formed that use a unary operation k times, a binary operation n-k times, and nothing else (sequence of operands is fixed). - Lars Hellstrom (Lars.Hellstrom(AT)residenset.net), Dec 08 2009

			Triangle begins:
00: [    1]
01: [    1,     1]
02: [    2,     3,      1]
03: [    5,    10,      6,      1]
04: [   14,    35,     30,     10,      1]
05: [   42,   126,    140,     70,     15,      1]
06: [  132,   462,    630,    420,    140,     21,     1]
07: [  429,  1716,   2772,   2310,   1050,    252,    28,    1]
08: [ 1430,  6435,  12012,  12012,   6930,   2310,   420,   36,   1]
09: [ 4862, 24310,  51480,  60060,  42042,  18018,  4620,  660,  45,  1]
10: [16796, 92378, 218790, 291720, 240240, 126126, 42042, 8580, 990, 55, 1]
...

Vincenzo Librandi, Rows n = 0..100, flattened
J. Agapito, A. Mestre, P. Petrullo, and M. Torres, Counting noncrossing partitions via Catalan triangles, CEAFEL Seminar, June 30, 2015.
Jean-Christophe Aval and François Bergeron, Rectangular Schröder Parking Functions Combinatorics, arXiv:1603.09487 [math.CO], 2016.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 25.
David Callan and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
G. E. Cossali, A Common Generating Function of Catalan Numbers and Other Integer Sequences, J. Int. Seqs. 6 (2003).
Dan Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) # 10.9.2
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.
Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.

Cf. A006318, A000108, A001700.
Triangle in A088617 transposed.
Diagonals give A000108, A001700, A002457, A002802, A002803; A000012, A000217, A034827, A000910, A088625, A088626.
T(2n,n) gives A007004.
Cf. A001263, A033282, A086810, A088617, A097610, A101282.

A060693 := (n,k) -> binomial(n,k)*binomial(2*n-k,n)/(n-k+1); # Peter Luschny, May 17 2011

t[n_, k_] := Binomial[n, k]*Binomial[2 n - k, n]/(n - k + 1); Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Robert G. Wilson v, May 30 2011 *)

T(n, k) = binomial(n, k)*binomial(2*n - k, n)/(n - k + 1);
for(n=0, 10, for(k=0, n, print1(T(n, k),", ")); print); \\ Indranil Ghosh, Jul 28 2017

from sympy import binomial
def T(n, k): return binomial(n, k) * binomial(2 * n - k, n) / (n - k + 1)
for n in range(11): print([T(n, k) for k in range(n + 1)])  # Indranil Ghosh, Jul 28 2017

Triangle T(n, k) (0 <= k <= n) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 12 2003
If C_n(x) is the g.f. of row n of the Narayana numbers (A001263), C_n(x) = Sum_{k=1..n} binomial(n,k-1)*(binomial(n-1,k-1)/k) * x^k and T_n(x) is the g.f. of row n of T(n,k), then T_n(x) = C_n(x+1), or T(n,k) = [x^n]Sum_{k=1..n}(A001263(n,k)*(x+1)^k). - Mitch Harris, Jan 16 2007, Jan 31 2007
G.f.: (1 - t*y - sqrt((1-y*t)^2 - 4*y)) / 2.
T(n, k) = binomial(2n-k, n)*binomial(n, k)/(n-k+1). - Philippe Deléham, Dec 07 2003
A060693(n, k) = binomial(2*n-k, k)*A000108(n-k); A000108: Catalan numbers. - Philippe Deléham, Dec 30 2003
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Apr 01 2007
T(n,k) = Sum_{j>=0} A090181(n,j)*binomial(j,k). - Philippe Deléham, May 04 2007
Sum_{k=0..n} T(n,k)*x^(n-k) = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Oct 18 2007
From Paul Barry, Jan 29 2009: (Start)
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-.... (continued fraction);
G.f.: 1/(1-(x+xy)/(1-x/(1-(x+xy)/(1-x/(1-(x+xy)/(1-.... (continued fraction). (End)
T(n,k) = [k<=n]*(Sum_{j=0..n} binomial(n,j)^2*binomial(j,k))/(n-k+1). - Paul Barry, May 28 2009
T(n,k) = A104684(n,k)/(n-k+1). - Peter Luschny, May 17 2011
From Tom Copeland, Sep 21 2011: (Start)
With F(x,t) = (1-(2+t)*x-sqrt(1-2*(2+t)*x+(t*x)^2))/(2*x) an o.g.f. (nulling the n=0 term) in x for the A060693 polynomials in t,
  G(x,t) = x/(1+t+(2+t)*x+x^2) is the compositional inverse in x.
Consequently, with H(x,t) = 1/(dG(x,t)/dx) = (1+t+(2+t)*x+x^2)^2 / (1+t-x^2), the n-th A060693 polynomial in t is given by (1/n!)*((H(x,t)*d/dx)^n) x evaluated at x=0, i.e., F(x,t) = exp(x*H(u,t)*d/d) u, evaluated at u = 0.
  Also, dF(x,t)/dx = H(F(x,t),t). (End)
See my 2008 formulas in A033282 to relate this entry to A088617, A001263, A086810, and other matrices. - Tom Copeland, Jan 22 2016
Rows of this entry are non-vanishing antidiagonals of A097610. See p. 14 of Agapito et al. for a bivariate generating function and its inverse. - Tom Copeland, Feb 03 2016
From Werner Schulte, Jan 09 2017: (Start)
T(n,k) = A126216(n,k-1) + A126216(n,k) for 0 < k < n;
Sum_{k=0..n} (-1)^k*(1+x*(n-k))*T(n,k) = x + (1-x)*A000007(n).
(End)
Conjecture: Sum_{k=0..n} (-1)^k*T(n,k)*(n+1-k)^2 = 1+n+n^2. - Werner Schulte, Jan 11 2017

More terms from Vladeta Jovovic, Apr 21 2001
New description from Philippe Deléham, Aug 12 2003
New name using a comment by Emeric Deutsch from Peter Luschny, Jul 26 2017

A107841 Series reversion of x(1-3x)/(1-x).

Original entry on oeis.org

1, 2, 10, 62, 430, 3194, 24850, 199910, 1649350, 13879538, 118669210, 1027945934, 9002083870, 79568077034, 708911026210, 6359857112438, 57403123415350, 520895417047010, 4749381474135850, 43489017531266654, 399755692955359630, 3687437532852484442, 34121911117572911410

Offset: 0

Paul Barry, May 24 2005

In general, the series reversion of x(1-r*x)/(1-x) has g.f. (1+x-sqrt(1+2*(1-2*r)*x+x^2))/(2*r) and general term given by a(n)=(1/(n+1))sum{k=0..n, C(n+1,k)C(2n-k,n)(-1)^k*r^(n-k)}; a(n)=(1/(n+1))sum{k=0..n, C(n+1,k+1)C(n+k,k)(-1)^(n-k)*r^k}; a(n)=sum{k=0..n, (1/(k+1))*C(n,k)C(n+k,k)(-1)^(n-k)*r^k}; a(n)=sum{k=0..n, A088617(n,k)*(-1)^(n-k)*r^k}.
The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 29 2007
Number of Dyck n-paths with three colors of up (U,a,b) and one color of down (D) avoiding UD. - David Scambler, Jun 24 2013
This sequence is implied in the turbulence solutions of the incompressible Navier-Stokes equations in R^3. a(n) = numbers of realizable vorticity eddies in terms of initial conditions. - Fung Lam, Dec 31 2013
Conjugate sequence to this series is defined by series reversion of x(1+3*x)/(1+x), G.f.: ((x-1)-sqrt(1-10*x+ x^2))/(6*x). Conjugate sequence is the negation of this series except a(0). - Fung Lam, Jan 16 2014
Complete Chebyshev transform is G.f. = 3*F((1-x^2)/(1+x^2)), where F(x) is the g.f. of A107841. Real part of G.f. (= (1 - sqrt(3*x^4-2))/((1+x^2))) generates periodic sequence A056594. In general, for reversion of x*(1-r*x)/(1-x), r>=2, Real part of r*F((1-x^2)/(1+x^2)) (= (1 - sqrt(r*x^4 - r + 1))/(1+x^2)) generates A056594. - Fung Lam, Apr 29 2014
a(n) is the number of small Schröder n-paths with 2 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 07 2019

Fung Lam, Table of n, a(n) for n = 0..1000 [The first 200 terms were computed by Vincenzo Librandi]
J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, p_n(2).
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016; eq. (1.13), a=2, b=3.
Samuele Giraudo, Operads from posets and Koszul duality, arXiv preprint arXiv:1504.04529 [math.CO], 2015.
F. Lam, Integral Invariance and Non-linearity Reduction for Proliferating Vorticity Scales in Fluid Dynamics, arXiv:1311.6395 [physics.flu-dyn], 2013-2014.
F. Lam, Vorticity evolution in a rigid pipe of circular cross-section, arXiv preprint arXiv:1505.07723 [physics.flu-dyn], 2015-2019.

Cf. A001003 (r=2), this sequence (r=3), A131763 (r=4), A131765 (r=5), A131846 (r=6), A131926 (r=7), A131869 (r=8), A131927 (r=9).

seq(simplify((-1)^n*hypergeom([-n, n + 1], [2], 3)), n=0..10); # Georg Fischer, Sep 14 2024 (from Peter Luschny's formula in A131763, with last parameter r=3)

CoefficientList[Series[(1+x-Sqrt[1-10*x+x^2])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

x='x+O('x^66); Vec(serreverse(x*(1-3*x)/(1-x))) \\ Joerg Arndt, May 15 2013

G.f.: (1+x-sqrt(1-10x+x^2))/(6x).
a(n) = (1/(n+1))sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*3^(n-k)}.
a(n) = (1/(n+1))sum{k=0..n, C(n+1, k+1)C(n+k, k)(-1)^(n-k)*3^k}.
a(n) = sum{k=0..n, (1/(k+1))*C(n, k)C(n+k, k)(-1)^(n-k)*3^k}.
a(n) = sum{k=0..n, A088617(n, k)*(-1)^(n-k)*3^k}.
a(n) = Sum_{k>=0} A086810(n, k)*2^k. - Philippe Deléham, May 26 2005
a(n) = (2/3)*A103210(n) for n>0. - Philippe Deléham, Oct 29 2007
G.f.: 1/(1-2x/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x........ (continued fraction). - Paul Barry, Dec 15 2008
From Paul Barry, May 15 2009: (Start)
G.f.: 1/(1-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-... (continued fraction).
G.f.: 1/(1-2x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-... (continued fraction). (End)
G.f.: 1/(1+x-3x/(1+x-3x/(1+x-3x/(1+x-3x/(1+x-3x/(1+... (continued fraction). - Paul Barry, Mar 18 2011
D-finite with recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
a(n+1) is the coefficient of x^(n+1) in 2*sum{j,1,n}((sum{k,1,n}a(k)x^k)^(j+1)), a(1)=1 with offset by 1. - Fung Lam, Dec 31 2013
The series reversion of x*(1 - r*x)/(1 - x) is D-finite with the general recurrence n*a(n) - (2*r-1)*(2*n-3)*a(n-1) + (n-3)*a(n-2) = 0 and with initial values a(1) = 1, a(2) = r-1, a(3) = (2*r-1)*(r-1). This sequence uses r=3, cf. crossrefs. - Georg Fischer, Sep 14 2024

A103209 Square array T(n,d) read by antidiagonals: number of structurally-different guillotine partitions of a d-dimensional box in R^d by n hyperplanes.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 22, 15, 4, 1, 90, 93, 28, 5, 1, 394, 645, 244, 45, 6, 1, 1806, 4791, 2380, 505, 66, 7, 1, 8558, 37275, 24868, 6345, 906, 91, 8, 1, 41586, 299865, 272188, 85405, 13926, 1477, 120, 9, 1, 206098, 2474025, 3080596, 1204245, 229326, 26845

Offset: 1

Ralf Stephan, Jan 27 2005

The columns are the row sums of the inverses of the Riordan arrays ((1-d*x)/(1-x),x(1-d*x)/(1-x)), that is, of the Riordan arrays ((1+x-sqrt(1+2(1-2*d)x+x^2)/(2*d*x),(1+x-sqrt(1+2(1-2*d)x+x^2)/(2*d)). - Paul Barry, May 24 2005

			1,...1,....1,.....1,......1,......1,.......1,.......1,.......1,
1,...2,....3,.....4,......5,......6,.......7,.......8,.......9,
1,...6,...15,....28,.....45,.....66,......91,.....120,.....153,
1,..22,...93,...244,....505,....906,....1477,....2248,....3249,
1,..90,..645,..2380,...6345,..13926,...26845,...47160,...77265,
1,.394,.4791,.24868,..85405,.229326,..522739,.1059976,.1968633,
1,1806,37275,272188,1204245,3956106,10663471,24958200,52546473,

E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett 98 (4) (2006) 162-167.
Andrei Asinowski and Toufik Mansour, Separable d-Permutations and Guillotine Partitions, arXiv 0803.3414 [math.CO], 2008; Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; Abstract
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Jean Cardinal, Stefan Felsner, Topological drawings of complete bipartite graphs, Journal of Computational Geometry 9.1 (2018), 213-246. Also arXiv:1608.08324 [cs.CG], 2016 (The OEIS is referenced in version v1 but not in v2).

Second column is A006318 (Schroeder numbers), others are A103210 and A103211. Main diagonal is A292798, diagonal under the main diagonal is A103212.

T := (n,k) -> hypergeom([-n, n+1], [2], -k);
seq(print(seq(simplify(T(n, k)), k=0..9)), n=0..6); # Peter Luschny, May 23 2014

T[0, ] = T[, 0] = 1;
T[n_, k_] := Sum[Binomial[n+j, 2j] k^j CatalanNumber[j], {j, 0, n}];
Table[T[n-k+1, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, after Paul Barry *)

T(n, d) = (1/n) * sum[i=0..n-1, C(n, i)*C(n, i+1)*(d-1)^i*d^(n-i) ], T(n, 0)=1.
G.f. of d-th column: [1-z-(z^2-4dz+2z+1)^(1/2)]/(2dz-2z).
T(n, k) = sum{j=0..n, C(n+j, 2j)*k^j*C(j)}, C(n) as in A000108. - Paul Barry, May 21 2005
T(n, k) = hypergeom([-n, n+1], [2], -k). - Peter Luschny, May 23 2014

A336574 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(kn+j+1,n)/(kn+j+1).

Original entry on oeis.org

1, 1, 3, 1, 3, 6, 1, 3, 15, 12, 1, 3, 24, 93, 24, 1, 3, 33, 255, 645, 48, 1, 3, 42, 498, 3102, 4791, 96, 1, 3, 51, 822, 8691, 40854, 37275, 192, 1, 3, 60, 1227, 18708, 164937, 566934, 299865, 384, 1, 3, 69, 1713, 34449, 464115, 3305868, 8164263, 2474025, 768

Offset: 0

Seiichi Manyama, Jul 26 2020

			Square array begins:
   1,    1,     1,      1,      1,       1, ...
   3,    3,     3,      3,      3,       3, ...
   6,   15,    24,     33,     42,      51, ...
  12,   93,   255,    498,    822,    1227, ...
  24,  645,  3102,   8691,  18708,   34449, ...
  48, 4791, 40854, 164937, 464115, 1055838, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A003945, A103210, A219536, A336539, A336572.
Main diagonal gives A336577.
Cf. A336534, A336573, A336575.

T[n_, k_] := Sum[2^j * Binomial[n, j] * Binomial[k*n + j + 1, n]/(k*n + j + 1), {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 27 2020 *)

T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));

T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k*(1+2*A)); polcoeff(A, n);

T(n, k) = sum(j=0, n, 2^(n-j)*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (1 + 2 * A_k(x)).
T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} 2^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
From Seiichi Manyama, Aug 10 2023: (Start)
T(n,k) = (1/n) * Sum_{j=0..n-1} (-1)^j * 3^(n-j) * binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0.
T(n,k) = (1/n) * Sum_{j=1..n} 3^j * 2^(n-j) * binomial(n,j) * binomial(k*n,j-1) for n > 0. (End)
T(n,k) = binomial(1+k*n, n)*hypergeom([-n, 1+k*n], [2+(k-1)*n], -2)/(1 + k*n) for k > 0. - Stefano Spezia, Aug 09 2025

A133305 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)4^i5^(n-i), a(0) = 1.

Original entry on oeis.org

1, 5, 45, 505, 6345, 85405, 1204245, 17558705, 262577745, 4005148405, 62070886845, 974612606505, 15471084667545, 247876665109005, 4003225107031845, 65101209768055905, 1065128963164067745, 17520376884067071205, 289572455530026439245, 4806489064223483202905

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Fifth column of array A103209.

The Hankel transform of this sequence is 20^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..795
Samuele Giraudo, Operads from posets and Koszul duality, arXiv:1504.04529 [math.CO], 2015-2016.

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-18*x+1))/(8*x))) // G. C. Greubel, Feb 10 2018

a[n_] := Hypergeometric2F1[-n, n + 1, 2, -4];
Table[a[n], {n, 0, 16}] (* Peter Luschny, Jan 08 2018 *)
CoefficientList[Series[(1-x-Sqrt[x^2-18*x+1])/(8*x), {x, 0, 50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-18*x+1))/(8*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-18*z+1))/(8*z).
a(n) = Sum_{k=0..n} A088617(n,k)*4^k.
a(n) = Sum_{k=0..n} A060693(n,k)*4^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)*4^k*C(k), C(n) given by A000108.
a(0) = 1, a(n) = a(n-1) + 4*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 9*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: 1/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) = hypergeom([-n, n + 1], [2], -4). - Peter Luschny, Jan 08 2018
a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 21 2021

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A166696 A transform of A103210.

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A166697 A "Morgan Voyce" transform of A103210.

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A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.

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A088617 Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.

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A060693 Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.

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A107841 Series reversion of x*(1-3*x)/(1-x).

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A107841 Series reversion of x(1-3x)/(1-x).

A336574 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(kn+j+1,n)/(kn+j+1).

A133305 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)4^i5^(n-i), a(0) = 1.