A007564 -id:A007564 - 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

A047656 a(n) = 3^((n^2-n)/2).

Original entry on oeis.org

1, 1, 3, 27, 729, 59049, 14348907, 10460353203, 22876792454961, 150094635296999121, 2954312706550833698643, 174449211009120179071170507, 30903154382632612361920641803529, 16423203268260658146231467800709255289, 26183890704263137277674192438430182020124347

Offset: 0

N. J. A. Sloane

The number of outcomes of a chess tournament with n players.
For n >= 1, a(n) is the size of the Sylow 3-subgroup of the Chevalley group A_n(3) (sequence A053290). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
The number of binary relations on an n-element set that are both reflexive and antisymmetric. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
The sequence a(n+1) = [1,3,27,729,59049,14348907,...] is the Hankel transform (see A001906 for definition) of A047891 = 1, 3, 12, 57, 300, 1586, 9912, ... . - Philippe Deléham, Aug 29 2006
a(n) is the number of binary relations on a set with n elements that are total relations, i.e., for a relation on a set X it holds for all a and b in X that a~b or b~a (or both). E.g., a(2) = 3 because there are three total relations on a set with two elements: {(a,a),(a,b),(b,a),(b,b)}, {(a,a),(a,b),(b,b)}, and {(a,a),(b,a),(b,b)}. - Geoffrey Critzer, May 23 2008
The number of semicomplete digraphs (or weak tournaments) on n labeled nodes. - Rémy-Robert Joseph, Nov 12 2012
The number of n X n binary matrices A that have a(i,j)=0 whenever a(j,i)=1 for i!=j and zeros on the diagonal. We need only consider the (n^2-n)/2 non-diagonal entry pairs . Since each pair is of the form <0,0>, <0,1>, or <1,0>, a(n) = 3^((n^2-n)/2). - Dennis P. Walsh, Apr 03 2014
a(n) is the number of symmetric (-1,0,1)-matrices of dimension (n-1) X (n-1). - Eric W. Weisstein, Jan 03 2021

			The a(2)=3 binary 2 X 2 matrices are [0 0; 0 0], [0 1; 0 0], and [0 0; 1 0]. - _Dennis P. Walsh_, Apr 03 2014

P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375.

Vincenzo Librandi, Table of n, a(n) for n = 0..65
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
T. R. Hoffman and J. P. Solazzo, Complex Two-Graphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334 [math.CO], 2014.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Eric Weisstein's World of Mathematics, (-1,0,1)-Matrix
Eric Weisstein's World of Mathematics, Symmetric Matrix
Index entries for sequences related to tournaments

Cf. A007747.

seq(3^binomial(n, 2), n=0..12); # Zerinvary Lajos, Jun 16 2007
seq(3^((n^2-n)/2), n=0..14);

Table[3^((n^2 - n)/2), {n, 0, 14}] (* Eric W. Weisstein, Jan 03 2021 *)
3^Table[Binomial[n, 2], {n, 0, 14}] (* Eric W. Weisstein, Jan 03 2021 *)
3^Binomial[Range[0, 14], 2] (* Eric W. Weisstein, Jan 03 2021 *)
Table[Count[Tuples[{-1, 0, 1}, {n, n}], ?SymmetricMatrixQ], {n, 3}] (* _Eric W. Weisstein, Jan 03 2021 *)

a(n)=3^binomial(n+1,2) \\ Charles R Greathouse IV, Apr 17 2012

a(n+1) is the determinant of an n X n matrix M_(i, j) = C(3*i,j). - Benoit Cloitre, Aug 27 2003
Sequence is given by the Hankel transform (see A001906 for definition) of A007564 = {1, 1, 4, 19, 100, 562, 3304, ...}; example: det([1, 1, 4, 19; 1, 4, 19, 100; 4, 19, 100, 562; 19, 100, 562, 3304]) = 3^6 = 729. - Philippe Deléham, Aug 20 2005
The sequence a(n+1) = [1,3,27,729,59049,14348907,...] is the Hankel transform (see A001906 for definition) of A047891 = 1, 3, 12, 57, 300, 1586, 9912, ... . - Philippe Deléham, Aug 29 2006
a(n) = 3^binomial(n,2). - Zerinvary Lajos, Jun 16 2007
G.f. A(x) satisfies: A(x) = 1 + x * A(3*x). - Ilya Gutkovskiy, Jun 04 2020
a(n) = a(n-1)*3^(n-1), a(0) = 1. - Mehdi Naima, Mar 09 2022

A059231 Number of different lattice paths running from (0,0) to (n,0) using steps from S = {(k,k) or (k,-k): k positive integer} that never go below the x-axis.

Original entry on oeis.org

1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405, 231153133, 1841801065, 14810069497, 120029657805, 979470140661, 8040831465825, 66361595715105, 550284185213925, 4582462506008253, 38306388126997785, 321327658068506121, 2703925940081270205

Offset: 0

Wenjin Woan, Jan 20 2001

If y = x*A(x) then 4*y^2 - (1 + 3*x)*y + x = 0 and x = y*(1 - 4*y) / (1 - 3*y). - Michael Somos, Sep 28 2003
a(n) = A059450(n, n). - Michael Somos, Mar 06 2004
The Hankel transform of this sequence is 4^binomial(n+1,2). - Philippe Deléham, Oct 29 2007
a(n) is the number of Schroder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 3 colors. Example: a(2)=5 because we have UDUD, UUDD, UBD, UGD, and URD, where U=(1,1), D=(1,-1), while B, G, and R are, respectively, blue, green, and red (2,0)-steps. - Emeric Deutsch, May 02 2011
Shifts left when INVERT transform applied four times. - Benedict W. J. Irwin, Feb 02 2016

			a(3) = 29 since the top row of Q^2 = (5, 8, 16, 0, 0, 0, ...), and 5 + 8 + 16 = 29.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.8.
Zhi Chen and Hao Pan, Identities involving weighted Catalan, Schroder and Motzkin paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=1, b=4.
Curtis Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28 (the sequence d_n).
Curtis Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
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.
Joseph P. S. Kung and Anna de Mier, Catalan lattice paths with rook, bishop and spider steps, J. Comb. Theor., Series A (2013) Vol. 120, Issue 2, 379-389.
Gregory J. Morrow, Laws relating runs and steps in gambler’s ruin, Stochastic Proc. Appl. (2024) Vol. 125, Issue 5, 2010-2025.
Gregory Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See pp. 1, 4, 15, 22. DOI
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 98.
Wen-jin Woan, Diagonal lattice paths, Congr. Numer. 151 (2001) 173-178.

Cf. A000108, A001003, A007564, A127846.
Row sums of A086873.
Column k=2 of A227578. - Alois P. Heinz, Jul 17 2013

gf := (1+3*x-sqrt(9*x^2-10*x+1))/(8*x): s := series(gf, x, 100): for i from 0 to 50 do printf(`%d,`,coeff(s, x, i)) od:
A059231_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+4*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A059231_list(20); # Peter Luschny, May 19 2011

Join[{1},Table[-I 3^n/2LegendreP[n,-1,5/3],{n,40}]] (* Harvey P. Dale, Jun 09 2011 *)
Table[Hypergeometric2F1[-n, 1 - n, 2, 4], {n, 0, 22}] (* Arkadiusz Wesolowski, Aug 13 2012 *)

{a(n) = if( n<0, 0, polcoeff( (1 + 3*x - sqrt(1 - 10*x + 9*x^2 + x^2 * O(x^n))) / (8*x), n))}; /* Michael Somos, Sep 28 2003 */

{a(n) = if( n<0, 0, n++; polcoeff( serreverse( x * (1 - 4*x) / (1 - 3*x) + x * O(x^n)), n))}; /* Michael Somos, Sep 28 2003 */

# Algorithm of L. Seidel (1877)
def A059231_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; b = False; h = 1
    for i in range(2*n) :
        if b :
            for k in range(1, h, 1) : D[k] += 2*D[k+1]
        else :
            for k in range(h, 0, -1) : D[k] += 2*D[k-1]
            h += 1
        b = not b
        if b : R.append(D[1])
    return R
A059231_list(23)  # Peter Luschny, Oct 19 2012

a(n) = Sum_{k=0..n} 4^k*N(n, k) where N(n, k) = (1/n)*binomial(n, k)*binomial(n, k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 10 2003
a(n) = 3^n/2*LegendreP(n, -1, 5/3). - Vladeta Jovovic, Sep 17 2003
G.f.: (1 + 3*x - sqrt(1 - 10*x + 9*x^2)) / (8*x) = 2 / (1 + 3*x + sqrt(1 - 10*x + 9*x^2)). - Michael Somos, Sep 28 2003
a(n) = Sum_{k=0..n} A088617(n, k)*4^k*(-3)^(n-k). - Philippe Deléham, Jan 21 2004
With offset 1: a(1)=1, a(n) = -3*a(n-1) + 4*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
D-finite with recurrence a(n) = (5(2n-1)a(n-1) - 9(n-2)a(n-2))/(n+1) for n>=2; a(0)=a(1)=1. - Emeric Deutsch, Mar 20 2004
Moment representation: a(n)=(1/(8*Pi))*Int(x^n*sqrt(-x^2+10x-9)/x,x,1,9)+(3/4)*0^n. - Paul Barry, Sep 30 2009
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  4, 4, 4
  1, 1, 1, 1
  4, 4, 4, 4, 4
  1, 1, 1, 1, 1, 1
  ... - Gary W. Adamson, Jul 08 2011
a(n) is the sum of top row terms of Q^(n-1), where Q = the following infinite square production matrix:
  1, 4, 0, 0, 0, ...
  1, 1, 4, 0, 0, ...
  1, 1, 1, 4, 0, ...
  1, 1, 1, 1, 4, ...
  ... - Gary W. Adamson, Aug 23 2011
G.f.: (1+3*x-sqrt(9*x^2-10*x+1))/(8*x)=(1+3*x -G(0))/(4*x) ; G(k)= 1+x*3-x*4/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
a(n) ~ sqrt(2)*3^(2*n+1)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2012
a(n) = A127846(n) for n>0. - Philippe Deléham, Apr 03 2013
0 = a(n)*(+81*a(n+1) - 225*a(n+2) + 36*a(n+3)) + a(n+1)*(+45*a(n+1) + 82*a(n+2) - 25*a(n+3)) + a(n+2)*(+5*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Aug 25 2014
G.f.: 1/(1 - x/(1 - 4*x/(1 - x/(1 - 4*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017

A047891 Number of planar rooted trees with n nodes and tricolored end nodes.

Original entry on oeis.org

1, 3, 12, 57, 300, 1686, 9912, 60213, 374988, 2381322, 15361896, 100389306, 663180024, 4421490924, 29712558576, 201046204173, 1368578002188, 9366084668802, 64403308499592, 444739795023054, 3082969991029800

Offset: 1

Louis Shapiro

Essentially the same as A025231.
Also number of lattice paths from (0,0) to (n-1,n-1), with steps (1,0),(0,1) and (1,1), that never rise above the line y=x and the steps (1,1) are colored red or blue. - Emeric Deutsch, May 28 2003
The Hankel transform (see A001906 for definition) of this sequence forms A049656(n+1) = [1, 3, 27, 729, 59049, 14348907, ...]. - Philippe Deléham, Aug 29 2006
With a(0)=0, this is the series reversion of x(1-x)/(1+2x). - Paul Barry, Oct 18 2009
Row sums of the Riordan matrix A121576. - Emanuele Munarini, May 18 2011

			G.f. = x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1686*x^6 + 9912*x^7 + ...

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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 and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO] (2016), eq. (1.13), a=3, b=1.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
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.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Eric Weisstein's MathWorld, Legendre Polynomial.
Index entries for sequences related to rooted trees

Cf. A006318, A121576, A054872.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-2*x-Sqrt(1-8*x+4*x^2))/(2*x))); // G. C. Greubel, Feb 10 2018

A047891_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A047891_list(20); # Peter Luschny, May 19 2011

CoefficientList[Series[(1-2x-Sqrt[1-8x+4x^2])/(2x),{x,0,100}],x] (* Emanuele Munarini, May 18 2011 *)
a[ n_] := SeriesCoefficient[(1 - 2 x - Sqrt[1 - 8 x + 4 x^2]) / 2, {x, 0, n}]; (* Michael Somos, Apr 10 2014 *)
Table[2^(n-1) (LegendreP[n, 2] - LegendreP[n-2, 2])/(2n-1), {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
Table[3 Hypergeometric2F1[1-n, 2-n, 2, 3] - 2 KroneckerDelta[n-1], {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

makelist(sum(binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k,k,0,n)/2,n,0,24); /* Emanuele Munarini, May 18 2011 */

a(n)=if(n<2,n==1,n--;sum(k=0,n,3^k*binomial(n,k)*binomial(n,k-1))/n)

x='x+O('x^100); Vec((1-2*x-sqrt(1-8*x+4*x^2))/2) \\ Altug Alkan, Nov 02 2015

G.f.: (1 - 2*x - sqrt(1 - 8*x + 4*x^2))/2.
For n>0, a(n+1) = (1/n)*Sum_{k=0..n} 3^k*C(n, k)*C(n, k-1) - Benoit Cloitre, May 10 2003
a(1)=1, a(n) = 2*a(n-1) + Sum_{i=1..(n-1)} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
The Hankel transform (see A001906 for definition) of this sequence form A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ...]. - Philippe Deléham, Aug 29 2006
2*a(n) = A054872(n+1). - Philippe Deléham, Aug 17 2007
From Paul Barry, Feb 01 2009: (Start)
G.f.: x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction);
a(n+1) = Sum_{k=0..n} C(n+k,2k)*2^(n-k)*A000108(k). (End)
G.f.: x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-... (continued fraction). - Paul Barry, Oct 18 2009
a(1) = 1, for n>=1, a(n+1) = 3*A007564(n). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009
From Emanuele Munarini, May 18 2011: (Start)
a(n+1) = (Sum_{k=0..n} binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k)/2.
D-finite with recurrence: (n+2)*(n+3)*a(n+3) - 6*(n+2)^2*a(n+2) - 12*(n)^2*a(n+1) + 8*n*(n-1)*a(n) = 0. (End)
G.f.: A(x) = (1-2*x-sqrt(4*x^2-8*x+1))/2 = 1 - G(0); G(k)= 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012
G.f.: x/W(0), where W(k)= k+1 - 2*x*(k+1) - x*(k+1)*(k+2)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
From Vladimir Reshetnikov, Nov 01 2015: (Start)
a(n) = 2^(n-1)*(LegendreP_n(2) - LegendreP_{n-2}(2))/(2n-1).
a(n) = 3*hypergeom([1-n,2-n], [2], 3) - 2*0^(n-1). (End)
a(n) = 2^(n-1)*hypergeom([1-n, n], [2], -1/2). - Peter Luschny, Nov 25 2020
a(n) ~ 3^(1/4) * (1 + sqrt(3))^(2*n - 1) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 31 2021
D-finite with recurrence n*a(n) +4*(-2*n+3)*a(n-1) +4*(n-3)*a(n-2)=0. - R. J. Mathar, Aug 01 2022

More terms from Christian G. Bower, Dec 11 1999

A078009 a(0)=1, for n>=1 a(n) = Sum_{k=0..n} 5^kN(n,k) where N(n,k) = C(n,k)C(n,k+1)/n are the Narayana numbers (A001263).

Original entry on oeis.org

1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076

Offset: 0

Benoit Cloitre, May 10 2003

nonn

More generally coefficients of (1 + m*x - sqrt(m^2*x^2 - (2*m+2)*x + 1) )/( 2*m*x ) are given by a(n) = Sum_{k=0..n} (m+1)^k * N(n,k).
a(n) is the series reversion of x*(1-5*x)/(1-4*x). a(n+1) is the series reversion of x/(1 + 6*x + 5*x^2). a(n+1) counts (6,5)-Motzkin paths of length n, where there are 6 colors available for the H(1,0) steps and 5 for the U(1,1) steps. - Paul Barry, May 19 2005
The Hankel transform of this sequence is 5^C(n+1,2). - Philippe Deléham, Oct 29 2007
a(n) is the number of Schröder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 4 colors. Example: a(2)=6 because we have UDUD, UUDD, UBD, UGD, URD, and UYD, where U=(1,1), D=(1,-1), while B, G, R, and Y are, respectively, blue, green, red, and yellow (2,0)-steps. - Emeric Deutsch, May 02 2011
Shifts left when INVERT transform applied five times. - Benedict W. J. Irwin, Feb 03 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Xiaomei Chen, Yuan Xiang, Counting generalized Schröder paths, arXiv:2009.04900 [math.CO], 2020.

Cf. A001003, A007564, A059231, A127848.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+4*x - Sqrt(16*x^2-12*x+1))/(10*x) )); // G. C. Greubel, Jun 28 2019

[1] cat [&+[5^k*Binomial(n,k)*Binomial(n,k+1)/n:k in [0..n]]:n in [1..20]]; // Marius A. Burtea, Jan 21 2020

A078009_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+5*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A078009_list(20); # Peter Luschny, May 19 2011

Table[SeriesCoefficient[(1+4*x-Sqrt[16*x^2-12*x+1])/(10*x),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 5];
Table[a[n], {n, 0, 30}] (* Peter Luschny, Mar 19 2018 *)

a(n)=sum(k=0,n,5^k/n*binomial(n,k)*binomial(n,k+1))

a=((1+4*x -sqrt(16*x^2-12*x+1))/(10*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 28 2019

G.f.: (1 + 4*x - sqrt(16*x^2 - 12*x + 1))/(10*x).
a(n) = Sum_{k=0..n} A088617(n, k)*5^k*(-4)^(n-k). - Philippe Deléham, Jan 21 2004
With offset 1 : a(1)=1, a(n) = -4*a(n-1) + 5*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
a(n+1) = Sum_{k=0..floor(n/2)} C(n, 2*k)*C(k)*6^(n-2k)*5^k; - Paul Barry, May 19 2005
a(n) = ( 6*(2*n-1)*a(n-1) - 16*(n-2)*a(n-2) ) / (n+1) for n >= 2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  5, 5, 5
  1, 1, 1, 1
  5, 5, 5, 5, 5
  1, 1, 1, 1, 1, 1
  ... (End)
a(n) ~ sqrt(10+6*sqrt(5))*(6+2*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012. Equivalently, a(n) ~ 2^(2*n) * phi^(2*n + 1) / (5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
a(n) = A127848(n) for n > 0. - Philippe Deléham, Apr 03 2013
G.f.: 1/(1 - x/(1 - 5*x/(1 - x/(1 - 5*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017
a(n) = hypergeom([1 - n, -n], [2], 5). - Peter Luschny, Mar 19 2018

A078018 a(n) = Sum_{k=0..n} 6^kN(n,k), with a(0)=1, where N(n,k) = C(n,k) C(n,k+1)/n are the Narayana numbers (A001263).

Original entry on oeis.org

1, 1, 7, 55, 469, 4237, 39907, 387739, 3858505, 39130777, 402972031, 4202705311, 44299426717, 471189693925, 5051001609115, 54513542257795, 591858123926545, 6459813793353265, 70837427884259575, 780073647992404615

Offset: 0

Benoit Cloitre, May 10 2003

nonn

More generally, coefficients of (1 + m*x - sqrt(m^2*x^2 - (2*m+4)*x + 1) )/( (2*m+2)*x ) are given by a(n) = Sum_{k=0..n} (m+1)^k*N(n,k).
The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 29 2007
Shifts left when INVERT transform applied six times. - Benedict W. J. Irwin, Feb 07 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A001003, A007564, A059231.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 + 5*x - Sqrt(25*x^2-14*x+1))/(12*x) )); // G. C. Greubel, Jun 29 2019

A078018_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+6*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A078018_list(19);
# Peter Luschny, May 19 2011

Table[SeriesCoefficient[(1+5*x-Sqrt[25*x^2-14*x+1])/(12*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_]:= Hypergeometric2F1[1 - n, -n, 2, 6]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Mar 19 2018 *)

a(n)=if(n<1,1,sum(k=0,n,6^k/n*binomial(n,k)*binomial(n,k+1)))

a=((1 + 5*x - sqrt(25*x^2-14*x+1))/(12*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 29 2019

G.f.: (1 + 5*x - sqrt(25*x^2-14*x+1))/(12*x).
a(n) = Sum_{k=0..n} A088617(n, k)*6^k*(-5)^(n-k). - Philippe Deléham, Jan 21 2004
a(n) = ( 7*(2*n-1)*a(n-1) - 25*(n-2)*a(n-2) ) / (n+1) for n>=2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1;
  6, 6, 6;
  1, 1, 1, 1;
  6, 6, 6, 6, 6;
  1, 1, 1, 1, 1, 1;
  ... (End)
a(n) ~ sqrt(12+7*sqrt(6))*(7+2*sqrt(6))^n/(12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
G.f.: 1/(1 - x/(1 - 6*x/(1 - x/(1 - 6*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017
a(n) = hypergeom([1 - n, -n], [2], 6). - Peter Luschny, Mar 19 2018

A008550 Table T(n,k), n>=0 and k>=0, read by antidiagonals: the k-th column given by the k-th Narayana polynomial.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 11, 4, 1, 1, 1, 42, 45, 19, 5, 1, 1, 1, 132, 197, 100, 29, 6, 1, 1, 1, 429, 903, 562, 185, 41, 7, 1, 1, 1, 1430, 4279, 3304, 1257, 306, 55, 8, 1, 1, 1, 4862, 20793, 20071, 8925, 2426, 469, 71, 9, 1, 1

Offset: 0

Philippe Deléham, Jan 23 2004

Mirror image of A243631. - Philippe Deléham, Sep 26 2014

			Row n=0:  1, 1,  1,   1,    1,     1,      1, ... see A000012.
Row n=1:  1, 1,  2,   5,   14,    42,    132, ... see A000108.
Row n=2:  1, 1,  3,  11,   45,   197,    903, ... see A001003.
Row n=3:  1, 1,  4,  19,  100,   562,   3304, ... see A007564.
Row n=4:  1, 1,  5,  29,  185,  1257,   8925, ... see A059231.
Row n=5:  1, 1,  6,  41,  306,  2426,  20076, ... see A078009.
Row n=6:  1, 1,  7,  55,  469,  4237,  39907, ... see A078018.
Row n=7:  1, 1,  8,  71,  680,  6882,  72528, ... see A081178.
Row n=8:  1, 1,  9,  89,  945, 10577, 123129, ... see A082147.
Row n=9:  1, 1, 10, 109, 1270, 15562, 198100, ... see A082181.
Row n=10: 1, 1, 11, 131,  161,  1661,  22101, ... see A082148.
Row n=11: 1, 1, 12, 155, 2124, 30482, 453432, ... see A082173.
... - _Philippe Deléham_, Apr 03 2013
The first few rows of the antidiagonal triangle are:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  1,  1;
  1,  5,  3,  1, 1;
  1, 14, 11,  4, 1, 1;
  1, 42, 45, 19, 5, 1, 1; - _G. C. Greubel_, Feb 15 2021

Michael De Vlieger, Table of n, a(n) for n = 0..11475
H. Prodinger, On a functional difference equation of Runyon, Morrison, Carlitz, and Riordan, arXiv:math/0103149 [math.CO], 2001.
H. Prodinger, On a functional difference equation of Runyon, Morrison, Carlitz, and Riordan, Séminaire Lotharingien de Combinatoire 46 (2001), Article B46a.
L. Yang, S.-L. Yang, A relation between Schroder paths and Motzkin paths, Graphs Combinat. 36 (2020) 1489-1502, eq. (6).

Columns: A000012, A000012, A000027, A028387, A090197, A090198, A090199, A090200.
Main diagonal is A242369.
A diagonal is in A099169.
Cf. A204057 (another version), A088617, A243631.
Cf. A132745.

[Truncate(HypergeometricSeries(k-n, k-n+1, 2, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 15 2021

gf := n -> 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1):
for n from 0 to 11 do PolynomialTools:-CoefficientList(convert( series(gf(n),x,12),polynom),x) od; # Peter Luschny, Nov 17 2014

(* First program *)
Unprotect[Power]; Power[0 | 0, 0 | 0] = 1; Protect[Power]; Table[Function[n, Sum[Apply[Binomial[#1 + #2, #1] Binomial[#1, #2]/(#2 + 1) &, {k, j}]*n^j*(1 - n)^(k - j), {j, 0, k}]][m - k + 1] /. k_ /; k <= 0 -> 1, {m, -1, 9}, {k, m + 1, 0, -1}] // Flatten (* Michael De Vlieger, Aug 10 2017 Note: this code renders 0^0 = 1. To restore normal Power functionality: Unprotect[Power]; ClearAll[Power]; Protect[Power] *)
(* Second program *)
Table[Hypergeometric2F1[1-n+k, k-n, 2, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)

flatten([[hypergeometric([k-n, k-n+1], [2], k).simplify_hypergeometric() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 15 2021

T(n, k) = Sum_{j>0} A001263(k, j)*n^(j-1); T(n, 0)=1.
T(n, k) = Sum_{j, 0<=j<=k} A088617(k, j)*n^j*(1-n)^(k-j).
The o.g.f. of row n is gf(n) = 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1). - Peter Luschny, Nov 17 2014
G.f. of row n: 1/(1 - x/(1 - n*x/(1 - x/(1 - n*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017
T(n, k) = Hypergeometric2F1([k-n, k-n+1], [2], k), as a number triangle. - G. C. Greubel, Feb 15 2021

A081178 a(0) = 1; for n>=1, a(n) = Sum_{k=0..n} 7^kN(n,k), where N(n,k)=(1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

Original entry on oeis.org

1, 1, 8, 71, 680, 6882, 72528, 788019, 8766248, 99362894, 1143498224, 13326176998, 156950554384, 1865210341828, 22338852956064, 269355965364459, 3267146912972328, 39837475762660374, 488032452193307568

Offset: 0

Benoit Cloitre, May 10 2003

nonn

More generally, coefficients of (1+m*x - sqrt(m^2*x^2-(2*m+4)*x+1) )/((2*m+2)*x) are given by: a(n) = Sum_{k=0..n} (m+1)^k*N(n,k).
The Hankel transform of this sequence is 7^C(n+1,2). - Philippe Deléham, Oct 29 2007
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  7, 7, 7
  1, 1, 1, 1
  7, 7, 7, 7, 7
  1, 1, 1, 1, 1, 1
  ...
(End)
Shifts left when INVERT transform applied seven times. - Benedict W. J. Irwin, Feb 07 2016
G.f.: 1/(1 - x/(1 - 7*x/(1 - x/(1 - 7*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A001003, A001263, A007564, A059231.

B:=Binomial;
A081178:= func< n | n eq 0 select 1 else (&+[7^k*B(n,k)*B(n,k+1): k in [0..n]])/n >;
[A081178(n): n in [0..40]]; // G. C. Greubel, Jan 15 2024

A081178_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+7*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A081178_list(18); # Peter Luschny, May 19 2011

Table[SeriesCoefficient[(1+6*x-Sqrt[36*x^2-16*x+1])/(14*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 7];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

a(n)=if(n<1,1,sum(k=0,n,7^k/n*binomial(n,k)*binomial(n,k+1)))

def A081178(n):
    b=binomial;
    if n==0: return 1
    else: return (1/n)*sum(7^k*b(n,k)*b(n,k+1) for k in range(n+1))
[A081178(n) for n in range(41)] # G. C. Greubel, Jan 15 2024

G.f.: (1+6*x-sqrt(36*x^2-16*x+1))/(14*x).
a(n) = (8*(2*n-1)*a(n-1) - 36*(n-2)*a(n-2))/(n+1) for n>=2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005
a(n) ~ sqrt(14+8*sqrt(7))*(8+2*sqrt(7))^n/(14*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
a(n) = hypergeom([1 - n, -n], [2], 7). - Peter Luschny, Mar 19 2018

cp's OEIS Frontend

A265435 Riordan array (1, x*f(x)) where f(x) is the g.f. of A007564.

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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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A047656 a(n) = 3^((n^2-n)/2).

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A059231 Number of different lattice paths running from (0,0) to (n,0) using steps from S = {(k,k) or (k,-k): k positive integer} that never go below the x-axis.

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A047891 Number of planar rooted trees with n nodes and tricolored end nodes.

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A078009 a(0)=1, for n>=1 a(n) = Sum_{k=0..n} 5^kN(n,k) where N(n,k) = C(n,k)C(n,k+1)/n are the Narayana numbers (A001263).

A078018 a(n) = Sum_{k=0..n} 6^kN(n,k), with a(0)=1, where N(n,k) = C(n,k) C(n,k+1)/n are the Narayana numbers (A001263).

A081178 a(0) = 1; for n>=1, a(n) = Sum_{k=0..n} 7^kN(n,k), where N(n,k)=(1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).