A072547 -id:A072547 - cp's OEIS frontend

A000957 Fine's sequence (or Fine numbers): number of relations of valence >= 1 on an n-set; also number of ordered rooted trees with n nodes having root of even degree.

0, 1, 0, 1, 2, 6, 18, 57, 186, 622, 2120, 7338, 25724, 91144, 325878, 1174281, 4260282, 15548694, 57048048, 210295326, 778483932, 2892818244, 10786724388, 40347919626, 151355847012, 569274150156, 2146336125648, 8110508473252, 30711521221376

Offset: 0

N. J. A. Sloane

Row-sum of signed Catalan triangle A009766. - Wouter Meeussen
There are two schools of thought about the best indexing for these numbers. Deutsch and Shapiro have a(4) = 6 whereas here a(5) = 6. The formulas given here use both labelings.
From D. G. Rogers, Oct 18 2005: (Start)
I notice that you have some other zero-one evaluations of binary bracketings (such as A055395). But if you have an operation # with 0#0 = 1#0 = 1, 0#1 = 1#1 = 0, and look at the number of bracketings of a string of n 0's that come out 0, you get another instance of the Fine numbers.
For Z = 1 + x(ZW + WW) = 1 + x CW and W = x(ZZ + ZW) = xZC. Hence Z = 1 + xxCCZ, the functional equational for the g.f. of the Fine numbers. Indeed, C = Z + W = Z + xCZ.
In terms of rooted planar trees with root of even degree, this says that of all rooted planar trees, some have root of even degree (Z) and some have root of odd degree (xCZ). (End)
Hankel transform of a(n+1) = [1,0,1,2,6,18,57,186,...] is A000012 = [1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
Starting with offset 3 = iterates of M * [1,0,0,0,...] where M = a tridiagonal matrix with [0,2,2,2,...] as the main diagonal and [1,1,1,...] as the super and subdiagonals. - Gary W. Adamson, Jan 09 2009
Starting with 1 and convolved with A068875 = the Catalan numbers with offset 1. - Gary W. Adamson, May 01 2009
For a relation to non-crossing partitions of the root system A_n, see A100754. - Tom Copeland, Oct 19 2014
From Tom Copeland, Nov 02 2014: (Start)
Let P(x) = x/(1+x) with comp. inverse Pinv(x) = x/(1-x) = -P[-x], and C(x) = [1-sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108, with inverse Cinv(x) = x * (1-x).
Fin(x) = P[C(x)] = C(x)/[1 + C(x)] is an o.g.f. for the Fine numbers, A000957 with inverse Fin^(-1)(x) = Cinv[Pinv(x)] = Cinv[-P(-x)] = (x-2x^2)/(1-x)^2, and Fin(Cinv(x)) = P(x).
Mot(x) = C[P(x)] = C[-Pinv(-x)] gives an o.g.f. for shifted A005043, the Motzkin or Riordan numbers with comp. inverse Mot^(-1)(x) = Pinv[Cinv(x)] = (x - x^2) / (1 - x + x^2) (cf. A057078).
BTC(x) = C[Pinv(x)] gives A007317, a binomial transform of the Catalan numbers, with BTC^(-1)(x) = P[Cinv(x)] = (x-x^2) / (1 + x - x^2).
Fib(x) = -Fin[Cinv(Cinv(-x))] = -P[Cinv(-x)] = x + 2 x^2 + 3 x^3 + 5 x^4 + ... = (x+x^2)/[1-x-x^2] is an o.g.f. for the shifted Fibonacci sequence A000045, so the comp. inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).
Generalizing to P(x,t) = x /(1 + t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f. for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t+1].
(End)
a(n+1) is the number of Dyck paths of semilength n avoiding UD at Level 0. For n = 3 the a(4) = 2 such Dyck paths are UUUDDD and UUDUDD. - Ran Pan, Sep 23 2015
For n >= 3, a(n) is the number of permutations pi of [n-2] such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - Colin Defant, Sep 16 2018
Named after the American scientist Terrence Leon Fine (1939-2021). - Amiram Eldar, Jun 08 2021

			G.f. = x + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 57*x^7 + 186*x^8 + 622*x^9 + 2120*x^10 + ...

Emeric Deutsch and Louis W. Shapiro, Seventeen Catalan identities, Bull. Instit. Combin. Applic., Vol. 31 (2001), pp. 31-38.
Ki Hang Kim, Douglas G. Rogers and Fred W. Roush, Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577-594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013). - N. J. A. Sloane, Jun 05 2012
Louis W. Shapiro and Carol J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1670 (first 201 terms from T. D. Noe)
Francesca Aicardi, Fuss-Catalan Triangles, arXiv:2310.07317 [math.CO], 2023.
Martin Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb., Vol. 12 (2005), R31.
Elena Barcucci, Elisa Pergola, Renzo Pinzani and Simone Rinaldi, ECO method and hill-free generalized Motzkin paths, Séminaire Lotharingien de Combinatoire, 46 (2001).
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, Vol. 18 (2011), #P178.
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 4.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Paul Barry, Series reversion with Jacobi and Thron continued fractions, arXiv:2107.14278 [math.NT], 2021.
Paul Barry, Moment sequences, transformations, and Spidernet graphs, arXiv:2307.00098 [math.CO], 2023.
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 6, 10.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
Rachael Boyd and Richard Hepworth, Combinatorics of injective words for Temperley-Lieb algebras, arXiv:2006.04261 [math.AT], 2020.
Naiomi Tuere Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.
Naiomi T. Cameron and Jillian E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19 (2016), #16.6.1.
Peter J. Cameron, Some sequences of integers, Discrete Math., Vol. 75, No. 1-3 (1989), pp. 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., Vol. 43 (1989), pp. 89-102.
Johann Cigler and Christian Krattenthaler, Hankel determinants of linear combinations of moments of orthogonal polynomials, arXiv:2003.01676 [math.CO], 2020.
Ari Cruz, Pamela E. Harris, Kimberly J. Harry, Jan Kretschmann, Matt McClinton, Alex Moon, John O. Museus, and Eric Redmon, On some discrete statistics of parking functions, arXiv:2312.16786 [math.CO], 2023. See p. 13.
Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.
Dennis E. Davenport, Louis W. Shapiro and Leon C. Woodson, A bijection between the triangulations of convex polygons and ordered trees, Integers, Vol. 20 (2020), Article #A8.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Emeric Deutsch and Louis W. Shapiro, A survey of the Fine numbers, Discrete Math., Vol. 241 (2001), pp. 241-265.
Filippo Disanto, Andrea Frosini and Simone Rinaldi and Renzo Pinzani, The Combinatorics of Convex Permutominoes, Southeast Asian Bulletin of Mathematics, Vol. 32 (2008), pp. 883-912.
R. R. X. Du, J. He, and X. Yun, Counting vertices in plane and k-ary trees with given outdegree, arXiv preprint arXiv:1501.07468 [math.CO], 2015.
Shalosh B. Ekhad, Mingjia Yang and Doron Zeilberger, Automated proofs of many conjectured recurrences in the OEIS made by R. J. Mathar, arXiv:1707.04654 [math.HO], 2017.
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
Terrence Fine, Extrapolation when very little is known about the source, Information and Control, Vol. 16 (1970), pp. 331-359.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 27.
Taras Goy and Mark Shattuck, Hessenberg-Toeplitz Matrix Determinants with Schröder and Fine Number Entries, arXiv:2303.10223 [math.CO], 2023.
Taras Goy and Mark Shattuck, Hessenberg-Toeplitz Matrix Determinants with Schröder and Fine Number Entries, Carpathian Math. Publ. 15 (2023), No. 2, 420-436.
Candice Jean-Louis and Asamoah Nkwanta, Some algebraic structure of the Riordan group, Linear Algebra and its Applications, Vol. 438, No. 5 (2013), pp. 2018-2035. - _N. J. A. Sloane_, Jan 03 2013
Sergey Kitaev, Anna de Mier, Marc Noy, On the number of self-dual rooted maps, European J. Combin., Vol. 35 (2014), pp. 377-387. MR3090510. See page 827.
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012. - From _N. J. A. Sloane_, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, Vol. 4 (2001), Article 01.1.5.
Toufik Mansour and Mark Shattuck, Chebyshev Polynomials and Statistics on a New Collection of Words in the Catalan Family, arXiv:1407.3516 [math.CO], 2014.
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
Asamoah Nkwanta and Akalu Tefera, Curious Relations and Identities Involving the Catalan Generating Function and Numbers, Journal of Integer Sequences, Vol. 16 (2013), Article 13.9.5.
Paul Peart and Wen-Jin Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), Article 00.2.1.
Paul Peart and Wen-Jin Woan, Dyck Paths With No Peaks at Height k, J. Integer Sequences, Vol. 4 (2001), Article 01.1.3.
Aaron Robertson, Dan Saracino and Doron Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
D. G. Rogers, Similarity relations on finite ordered sets, J. Combin. Theory, Series A, Vol. 23, No. 1 (1977), pp. 88-98. Erratum, loc. cit., Vol. 25 (1978), pp. 95-96.
Louis W. Shapiro and Carol J. Wang, A bijection between 3-Motzkin paths and Schroder paths with no peak at odd height, JIS, Vol. 12 (2009), Article 09.3.2.
Volker Strehl, A note on similarity relations, Discr. Math., Vol. 19, No. 1 (1977), pp. 99-102.
Hua Sun and Yi Wang, A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers, J. Int. Seq., Vol. 17 (2014), Article 14.5.2
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, M.Sc. Thesis, Reykjavik Univ., May 2016.
Index entries for sequences related to rooted trees

A column of A065600.
Sequence with signs: A064310.
Bisections: A138413, A138414.
Logarithmic derivative: A072547.
Cf. A068875, A000108, A000045, A005043, A057078, A007317, A091867, A104597.
Cf. A000012, A009766, A039598, A055395, A064306, A100754, A126093.

a000957 n = a000957_list !! n
a000957_list = 0 : 1 :
   (map (`div` 2) $ tail $ zipWith (-) a000108_list a000957_list)
-- Reinhard Zumkeller, Nov 12 2011

[0,1] cat  [n le 1 select n-1 else (Catalan(n)-Self(n-1))/2: n in [1..30]]; // Vincenzo Librandi, Nov 17 2016

t1 := (1-sqrt(1-4*x))/(3-sqrt(1-4*x)); t2 := series(t1,x,90); A000957 := n- coeff(t2,x,n);
A000957 := proc(n): if n = 0 then 0 else add((-1)^(n+k-1)*binomial(n+k-1, n-1)*(n-k)/n, k=0..n-1) fi: end: seq(A000957(n), n=0..28); # Johannes W. Meijer, Jul 22 2013
# third Maple program:
a:= proc(n) option remember; `if`(n<3, n*(2-n),
      ((7*n-12)*a(n-1)+(4*n-6)*a(n-2))/(2*n))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Apr 23 2020

Table[ Plus@@Table[ (-1)^(m+n) (n+m)!/n!/m! (n-m+1)/(n+1), {m, 0, n} ], {n, 0, 36} ] (* Wouter Meeussen *)
a[0] = 0; a[n_] := (1/2)*(-3*(-1/2)^n + 2^(n+1)*(2n-1)!!* Hypergeometric2F1Regularized[2, 2n+1, n+2, -1]); (* Jean-François Alcover, Feb 22 2012 *)
Table[2^n (n-2) (2n-1)!! (3 (n-1) Hypergeometric2F1[1, 3-n, 3+n, 2] - n - 2)/(n+2)! + KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)

C(n):=binomial(2*n,n)/(n+1);
a(n):=if n<=0 then 0 else if n=1 then 1 else  sum(C(n-i-1)*(a(i)+a(i-1)),i,2,n-1);
/* Vladimir Kruchinin, Apr 23 2020 */

{a(n) = if( n<1, 0, polcoeff( 1 / (1 + 2 / (1 - sqrt(1 - 4*x + x*O(x^n)))), n))}; /* Michael Somos, Sep 17 2006 */

{a(n) = if( n<1, 0, polcoeff( 1 / (1 + 1 / serreverse(x - x^2 + x*O(x^n))), n))}; /* Michael Somos, Sep 30 2006 */

from itertools import count, islice
def A000957_gen(): # generator of terms
    yield from (0,1,0)
    a, c = 0, 1
    for n in count(1):
        yield (a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)
A000957_list = list(islice(A000957_gen(),20)) # Chai Wah Wu, Apr 26 2023

def Fine():
    f, c, n = 1, 1, 1
    yield 0
    while True:
        yield f
        n += 1
        c = c * (4*n - 6) // n
        f = (c - f) // 2
a = Fine()
print([next(a) for  in range(29)])  # _Peter Luschny, Nov 30 2016

Catalan(n) = 2*a(n+1) + a(n), n >= 1. [Corrected by Pontus von Brömssen, Jul 23 2022]
a(n) = (A064306(n-1) + (-1)^(n-1))/2^n, n >= 1.
G.f.: (1-sqrt(1-4*x))/(3-sqrt(1-4*x)) (compare g.f. for Catalan numbers, A000108). - Emeric Deutsch
a(n) ~ 4^n/(9*n*sqrt(n*Pi)). (Corrected by Peter Luschny, Oct 26 2015.)
a(n) = (2/(n-1))*Sum_{j=0..n-3}(-2)^j*(j+1)*binomial(2n-1, n-3-j), n>=2. - Emeric Deutsch, Dec 26 2003
a(n) = 3*Sum_{j=0..floor((n-1)/2)} binomial(2n-2j-2, n-1) - binomial(2n, n) for n>0. - Emeric Deutsch, Jan 28 2004
Reversion of g.f. (x-2x^2)/(1-x)^2. - Ralf Stephan, Mar 22 2004
a(n) = ((-1)^n/2^n)*(-3/4-(1/4)*sum{k=0..n, C(1/2, k)8^k})+0^n; a(n) = ((-1)^n/2^n)*(-3/4-(1/4)*sum{k=0..n, (-1)^(k-1)*2^k*(2k)!/((k!)^2*(2k-1))})+0^n. - Paul Barry, Jun 10 2005
Hankel determinant transform is 1-n. - Michael Somos, Sep 17 2006
a(n+1) = A126093(n,0). - Philippe Deléham, Mar 05 2007
a(n+1) has g.f. 1/(1-0*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(..... (continued fraction). - Paul Barry, Dec 02 2008
From Paul Barry, Jan 17 2009: (Start)
G.f.: x*c(x)/(1+x*c(x)), c(x) the g.f. of A000108;
a(n+1) = Sum_{k=0..n} (-1)^k*C(2n-k,n-k)*(k+1)/(n+1). (End)
a(n) = 3*(-1/2)^(n+1) + Gamma(n+1/2)*4^n*hypergeom([1, n+1/2],[n+2],-8) /(sqrt(Pi)*(n+1)!) (for n>0). - Mark van Hoeij, Nov 11 2009
Let A be the Toeplitz matrix of order n defined by: A[i,i-1] = -1, A[i,j] = Catalan(j-i), (i<=j), and A[i,j] = 0, otherwise. Then, for n>=1, a(n+1) = (-1)^n*charpoly(A,1). - Milan Janjic, Jul 08 2010
a(n) = the upper left term in M^n, n>0; where M = the infinite square production matrix:
0, 1, 0, 0, 0, 0, ...
1, 1, 1, 0, 0, 0, ...
1, 1, 1, 1, 0, 0, ...
1, 1, 1, 1, 1, 0, ...
1, 1, 1, 1, 1, 1, ...
...
- Gary W. Adamson, Jul 14 2011
a(n+1) = Sum_{k=0..n} A039598(n,k)*(-2)^k. - Philippe Deléham, Nov 04 2011
D-finite with recurrence: 2*n*a(n) +(12-7*n)*a(n-1) +2*(3-2*n)*a(n-2)=0. - R. J. Mathar, Nov 15 2011
a(n) = sum(sum(2^(s-2n-2k)*(n/n+2k)binomial(n+2k, k)*binomial(s-n-1, s-2n-2k), (k=0, ..., floor((s-2n)/2)), (n=1, ..., s) with s>=2. - José Luis Ramírez Ramírez, Mar 22 2012
0 = a(n)*(16*a(n+1) + 22*a(n+2) - 20*a(n+3)) + a(n+1)*(34*a(n+1) + 53*a(n+2) - 38*a(n+3)) + a(n+2)*(10*a(n+2) + 4*a(n+3)) for all n in Z if we extend by a(0)=-1, a(-n) = -3/4 * (-2)^n if n>0. - Michael Somos, Jan 31 2014 [Corrected by Pontus von Brömssen, Aug 04 2022]
G.f. A(x) satisfies x*A'(x)/A(x) = x + 2*x^3 + 6*x^4 + 22*x^5 + ..., the o.g.f. for A072547. - Peter Bala, Oct 01 2015
a(n) = 2^n*(n-2)*(2*n-1)!!*(3*(n-1)*hypergeom([1,3-n], [3+n], 2)-n-2)/(n+2)! + 0^n. - Vladimir Reshetnikov, Oct 25 2015
a(n) = binomial(2*n,n)*(hypergeom([1,(1-n)/2,1-n/2],[1-n,3/2-n],1)*3/(4-2/n)-1) for n>=2. - Peter Luschny, Oct 26 2015
O.g.f. A(x) satisfies 1 + A(x) = (1 + 3*Sum_{n >= 1} Catalan(n)*x^n)/(1 + 2*Sum_{n >= 1} Catalan(n)*x^n) = (1 + 2*Sum_{n >= 1} binomial(2*n,n)*x^n )/(1 + 3/2*Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016
a(n) = Sum_{i=2..n-1} C(n-i-1)*(a(i)+a(i-1)), a(0)=0, a(1)=1, where C(n) = A000108(n). - Vladimir Kruchinin, Apr 23 2020

A026641 Number of nodes of even outdegree (including leaves) in all ordered trees with n edges.

Original entry on oeis.org

1, 1, 4, 13, 46, 166, 610, 2269, 8518, 32206, 122464, 467842, 1794196, 6903352, 26635774, 103020253, 399300166, 1550554582, 6031074184, 23493410758, 91638191236, 357874310212, 1399137067684, 5475504511858, 21447950506396

Offset: 0

Clark Kimberling

Number of lattice paths from (0,0) to (n,n) using steps (1,0),(0,2),(1,1). - Joerg Arndt, Jun 30 2011
From Emeric Deutsch, Jan 25 2004: (Start)
Let B = 1/sqrt(1-4*z) = g.f. for central binomial coeffs (A000984); F = (1-sqrt(1-4*z))/(z*(3-sqrt(1-4*z))) = g.f. for (A000957).
B = 1 + 2*z + 6*z^2 + 20*z^3 + ... gives the number of nodes in all ordered trees with 0,1,2,3,... edges. On p. 288 of the Deutsch-Shapiro paper one finds that z*B*F = z + 2*z^2 + 7*z^3 + 24*z^4 + ... gives the number of nodes of odd outdegree in all ordered trees with 1,2,3,... edges (cf. A014300).
Consequently, B - z*B*F = 2/(3*sqrt(1-4*z)-1+4*z) = 1 + z + 4*z^2 + 13*z^3 + 46*z^4 + ... gives the total number of nodes of even degree in all ordered trees with 0,1,2,3,4,... edges.  (End)
Main diagonal of the following array: first column is filled with 1's, first row is filled alternatively with 1's or 0's: m(i,j) = m(i-1,j) + m(i,j-1): 1 0 1 0 1 ... / 1 1 2 2 3 ... / 1 2 4 6 9 ... / 1 3 7 13 22 ... / 1 4 11 24 46 ... - Benoit Cloitre, Aug 05 2002
The Hankel transform of [1,1,4,13,46,166,610,2269,...] is 3^n. - Philippe Deléham, Mar 08 2007
Second binomial transform of A127361. - Philippe Deléham, Mar 14 2007
Starting with offset 1, generated from iterates of M * [1,1,1,...]; where M = a tridiagonal matrix with (0,2,2,2,...) in the main diagonal and (1,1,1,...) in the super and subdiagonals. - Gary W. Adamson, Jan 04 2009
Equals left border of triangle A158815. - Gary W. Adamson, Mar 27 2009
Equals the INVERTi transform of A101850: (1, 2, 7, 26, 100, ...). - Gary W. Adamson, Jan 10 2012
Diagonal of rational function 1/(1 - (x + x*y + y^2)). - Gheorghe Coserea, Aug 06 2018
Let A(i, j) denote the infinite array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i times to the function (-1)^(n+1) for n > 0. Then A(n, n) equals a(n-1) for all n > 0. - John M. Campbell, Jan 20 2019
These numbers have the same parity as the Catalan numbers A000108; that is, a(n) is odd if and only if n = 2^k - 1 for some nonnegative integer k. It appears that if a(n) is odd then a(n) == 1 (mod 4). - Peter Bala, Feb 07 2024
The number a(n)/(n+1) is the coefficient of x^(n+1) in log(1+(1-sqrt(1-4*x))/2), the generating series of the Sabinin operad. - F. Chapoton, Mar 14 2024

			From _Joerg Arndt_, Jul 01 2011: (Start)
The triangle of number of lattice paths from (0,0) to (n,k) using steps (1,0),(0,2),(1,1) begins
  1;
  1, 1;
  1, 2,  4;
  1, 3,  7, 13;
  1, 4, 11, 24,  46;
  1, 5, 16, 40,  86, 166;
  1, 6, 22, 62, 148, 314,  610;
  1, 7, 29, 91, 239, 553, 1163, 2269;
This sequence is the diagonal. (End)
G.f. = 1 + x + 4*x^2 + 13*x^3 + 46*x^4 + 166*x^5 + 610*x^6 + 2269*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Karl Dilcher and Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1) = 1, arXiv:1909.11222 [math.NT], 2019.
Filippo Disanto, Andrea Frosini and Simone Rinaldi, Renzo Pinzani, The Combinatorics of Convex Permutominoes, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.

Cf. A101850, A120588, A158815.

Cf. A091526 (k=-2), A072547 (k=-1), this sequence (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172061 (k=4), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10).

List([0..25],n->(-1)^n*Sum([0..n],k->(-1)^k*Binomial(n+k,k))); # Muniru A Asiru, Aug 06 2018

[(-1)^n*(&+[(-1)^k*Binomial(n+k, k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 12 2019

seq(add((binomial(k+n, n-k)*binomial(n-k, k)),k=0..floor(n/2)),n=0..30);
# From Richard Choulet, Jan 22 2010: (Start)
a:= n -> add(binomial(2*n-k, k)*binomial(k, n-k), k=floor(n/2)..n):
a:= n -> `if`(n<2, 1, (3/(2))*binomial(2*n-1, n-1)-(1/2)*a(n-1)):
a:= n -> (-1/2)^(n+2)+(2/3)*add(4^(n-k)*(binomial(2*k, k)*(1/(1-2*k))
        *(1-(-1/8)^(n-k+1))), k=0..n):
a:= n -> (-1/2)^(n+2)+(3/4)*add(((-1/2)^(n-k))*(binomial(2*k, k)), k=0..n):
seq(a(n), n=0..30); # (End)
gf := log(1 + (1 - sqrt(1 - 4*x))/2) / x: ser := series(gf, x, 30):
seq((n + 1)*coeff(ser, x, n), n = 0..24);  # Peter Luschny, Mar 16 2024

f[n_]:= Sum[ Binomial[n+k, k]*Cos[Pi*(n+k)], {k, 0, n}]; Array[f, 25, 0] (* Robert G. Wilson v, Apr 02 2012 *)
CoefficientList[Series[2/(3*Sqrt[1-4*x]-1+4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
a[ n_]:= SeriesCoefficient[ D[ Log[1+(1-Sqrt[1-4x])/2], x], {x, 0, n}]; (* Michael Somos, May 18 2015 *)

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

/* same as in A092566 but use */
steps=[[1,0], [0,2], [1,1]]; /* Joerg Arndt, Jun 30 2011 */

[(-1)^n*sum((-1)^k*binomial(n+k, k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Feb 12 2019

G.f. is logarithmic derivative of the generating function for the Catalan numbers A000108. So this sequence might be called the "log-Catalan" numbers. - Murray R. Bremner, Jan 25 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(k+n, n-k)*binomial(n-k, k). - Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001
G.f.: 2/(3*sqrt(1-4*z)-1+4*z). - Emeric Deutsch, Jul 09 2002
a(n) = (-1)^n*Sum_{k=0..n} (-1)^k*C(n+k, k). - Benoit Cloitre, Aug 20 2002
a(n) = Sum_{j=0..floor(n/2)} binomial(2*n-2*j-1, n-1). - Emeric Deutsch, Jan 28 2004
From Paul Barry, Dec 18 2004: (Start)
A Catalan transform of the Jacobsthal numbers A001045(n+1) under the mapping G(x)-> G(xc(x)), c(x) the g.f. of A000108. The inverse mapping is H(x)->H(x(1-x)).
a(n) = Sum_{k=0..n} (k/(2*n-k))*binomial(2*n-k, n-k)*A001045(k+1). (End)
a(n) = Sum_{k=0..n} binomial(2*n-k, k)*binomial(k, n-k). - Paul Barry, Jul 25 2005
a(n) = Sum_{k=0..n-1} A126093(n,k). - Philippe Deléham, Mar 08 2007
a(n) = (-1/2)^(n+2) + (2/3)*Sum_{k=0..n} ( (4^n-k)*binomial(2*k,k)*(1/(1-2*k))*(1-(-1/8)^(n-k+1)) ). - Yalcin Aktar, Jul 06 2007
a(n) = (-1/2)^(n+2) + (3/4)*Sum_{k=0..n} (-1/2)^(n-k)*binomial(2*k,k). - Yalcin Aktar, Jul 06 2007
From Richard Choulet, Jan 22 2010: (Start)
a(n) = (3*binomial(2*n-1,n-1) - d(n-1))/2, where d(n) = Sum_{k=floor(n/2)..n} binomial(2*n-k, k)*binomial(k, n-k).
a(n) = a(n-1) + (3/2)*Sum_{k=2..n} (1/(2*k-1))*binomial(2*k,k)*a(n-k).
a(n) = (2/3)*binomial(2*n,n) + (2/9)*((-2)^n/n!)*Sum_{k>=0} ( Product_{p=0..n-1} (k-2*p) /3^k).
a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n-k,n).
a(n) = ( Sum_{k=0..n} (1/2)^(n-k+1)*binomial(n+k,k) )^2*(-1/2)^(n+2). (End)
From Gary W. Adamson, Nov 22 2011: (Start)
a(n) is the upper left term of M^n, M = an infinite square production matrix as follows:
  1, 3, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, ...
  ...
Also, a(n+1) is the sum of top row terms of M^n; e.g. top row of M^3 = (13, 21, 9, 3), sum = 46 = a(4), a(3) = 13. (End)
D-finite with recurrence: 2n*a(n) + (4-7n)*a(n-1) + 2*(1-2n)*a(n-2) = 0. - R. J. Mathar, Dec 17 2011 [The recurrence is proved with the Wilf-Zeilberger (WZ) method applied to Sum_{k=0..floor(n/2)} binomial(k+n, n-k)*binomial(n-k, k). - T. Amdeberhan, Jul 23 2012]
a(n) = A035317(2*n-1,n) for n > 0. - Reinhard Zumkeller, Jul 19 2012
a(n) ~ 2^(2*n+1) / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014
a(n) = binomial(2*n,n)*hypergeom([1, -n], [-2*n], -1). - Peter Luschny, May 22 2014
G.f. is the derivative of the logarithm of the g.f. for A120588. - Michael Somos, May 18 2015
a(n) = [x^n] 1/((1 - x^2)*(1 - x)^n). - Ilya Gutkovskiy, Oct 25 2017
From Peter Bala, Feb 25 2019: (Start)
a(n) = Sum_{k = 0..n} binomial(2*n + 1, n + k + 1)*(-2)^k.
a(n-1) = (1/2)*binomial(2*n,n)*( 1 - 2*(n-1)/(n+1) + 4*(n-1)*(n-2)/((n+1)*(n+2)) - 8*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ...) = (1/2)*binomial(2*n,n)*hypergeom([1 - n, 1], [n + 1], 2). (End)
a(0)=1, a(1)=1, and a(n) = (2 - 1/n)*a(n-2) + (7/2 - 2/n)*a(n-1) for n > 1. - Reginald Robson, Nov 01 2022

A112555 Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I) and consequently the matrix logarithm satisfies log(T) = T - I, where I is the identity matrix.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -2, 0, 1, 1, 3, 4, 2, 1, 1, -1, -4, -7, -6, -3, 0, 1, 1, 5, 11, 13, 9, 3, 1, 1, -1, -6, -16, -24, -22, -12, -4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, -1, -8, -29, -62, -86, -80, -50, -20, -5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, -1, -10, -46, -128, -239, -314, -296, -200, -95, -30, -6, 0

Offset: 0

Paul D. Hanna, Sep 21 2005

Signed version of A108561. Row sums equal A084247. The n-th unsigned row sum = A001045(n) + 1 (Jacobsthal numbers). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms.
Equals row reversal of triangle A112468 up to sign, where A112468 is the Riordan array (1/(1-x),x/(1+x)). - Paul D. Hanna, Jan 20 2006
The elements here match A108561 in absolute value, but the signs are crucial to the properties that the matrix A112555 exhibits; the main property being T^m = I + m*(T - I). This property is not satisfied by A108561. - Paul D. Hanna, Nov 10 2009
Eigensequence of the triangle = A140165. - Gary W. Adamson, Jan 30 2009
Triangle T(n,k), read by rows, given by [1,-2,0,0,0,0,0,0,0,...] DELTA [1,0,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 17 2009

			Triangle T begins:
   1;
   1,   1;
  -1,   0,   1;
   1,   1,   1,   1;
  -1,  -2,  -2,   0,   1;
   1,   3,   4,   2,   1,   1;
  -1,  -4,  -7,  -6,  -3,   0,   1;
   1,   5,  11,  13,   9,   3,   1,   1;
  -1,  -6, -16, -24, -22, -12,  -4,   0,   1;
   1,   7,  22,  40,  46,  34,  16,   4,   1,   1;
  -1,  -8, -29, -62, -86, -80, -50, -20,  -5,   0,   1;
  ...
Matrix log, log(T) = T - I, begins:
   0;
   1,  0;
  -1,  0,  0;
   1,  1,  1,  0;
  -1, -2, -2,  0,  0;
   1,  3,  4,  2,  1,  0;
  -1, -4, -7, -6, -3,  0,  0;
  ...
Matrix inverse, T^-1 = 2*I - T, begins:
   1;
  -1,  1;
   1,  0,  1;
  -1, -1, -1,  1;
   1,  2,  2,  0,  1;
  -1, -3, -4, -2, -1,  1;
  ...
where adjacent sums in row n of T^-1 gives row n+1 of T.

Paul D. Hanna, Table of n, a(n) for n = 0..1080

Cf. A112468 (reversed rows), A108561, A084247, A001045, A072547, A112556, A279006, A140165.
From Philippe Deléham, Oct 07 2009: (Start)
Sum_{k=0..n} T(n, k)*x^(n-k) = A165760(n), A165759(n), A165758(n), A165755(n), A165752(n), A165746(n), A165751(n), A165747(n), A000007(n), A000012(n), A084247(n), A165553(n), A165622(n), A165625(n), A165638(n), A165639(n), A165748(n), A165749(n), A165750(n) for x= -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.
Sum_{k=0..n} T(n, k)*x^k = A166157(n), A166153(n), A166152(n), A166149(n), A166036(n), A166035(n), A091004(n+1), A077925(n), A000007(n), A165326(n), A084247(n), A165405(n), A165458(n), A165470(n), A165491(n), A165505(n), A165506(n), A165510(n), A165511(n) for x = -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.  (End)

Clear[t]; t[0, 0] = 1; t[n_, 0] = (-1)^(Mod[n, 2]+1); t[n_, n_] = 1; t[n_, k_] /; k == n-1 := t[n, k] = Mod[n, 2]; t[n_, k_] /; 0 < k < n-1 := t[n, k] = -t[n-1, k] - t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)

{T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff( polcoeff( (1+2*x+x*y)/((1-x*y)*(1+x+x*y)),n,X),k,Y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

{T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff(1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)),n,X),k,Y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

def A112555_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k+1)*prec(n+1, k) for k in (1..n+1)]
for n in (0..12): print(A112555_row(n)) # Peter Luschny, Mar 16 2016

G.f.: 1/(1-x*y) + x/((1-x*y)*(1+x+x*y)).
The m-th matrix power T^m has the g.f.: 1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)).
Recurrence: T(n, k) = [T^-1](n-1, k) + [T^-1](n-1, k-1), where T^-1 is the matrix inverse of T.
From Peter Bala, Jun 23 2025: (Start)
T^z = exp(z*log(T)) = I + z*(T - I) for arbitrary complex z, where I is the identity array.
exp(T) = e*T. More generally, exp(z * T^u) = exp(z)*T^(u*z) = exp(z)*I + u*z*exp(z)*(T - I).
sin(z * T^u) =  sin(z)*I + u*z*cos(z)*(T - I).
cos(z * T^u) =  cos(z)*I - u*z*sin(z)*(T - I).
tan(z * T^u) =  tan(z)*I + u*z*sec(z)^2*(T - I).
Chebyshev_T(n, T^u) = I + (n^2)*u*(T - I) and
Legendre_P(n, T^u) = I + (n*(n+1)/2)*u*(T - I).
More generally, for n >= 1,
Chebyshev_T(n, z*T^u) = Chebyshev_T(n, z)*I + n*u*z*Chebyshev_U(n-1, z)*(T - I) and
Legendre_P(n, z*T^u) = Legendre_P(n, z)*I + u*Q(n, z)*(T - I), where Q(1, z) = z and Q(n, z) = n*Legendre_P(n, z) + Q(n-1, z)/z for n > 1.
All the above properties may also hold for the triangle A279006. (End)

A112468 Riordan array (1/(1-x), x/(1+x)).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 0, 2, -2, 1, 1, 1, -2, 4, -3, 1, 1, 0, 3, -6, 7, -4, 1, 1, 1, -3, 9, -13, 11, -5, 1, 1, 0, 4, -12, 22, -24, 16, -6, 1, 1, 1, -4, 16, -34, 46, -40, 22, -7, 1, 1, 0, 5, -20, 50, -80, 86, -62, 29, -8, 1, 1, 1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 1, 0, 6, -30, 95, -200, 296, -314, 239, -128, 46, -10, 1

Offset: 0

Paul Barry, Sep 06 2005

Row sums are A040000. Diagonal sums are A112469. Inverse is A112467. Row sums of k-th power are 1, k+1, k+1, k+1, .... Note that C(n,k) = Sum_{j=0..n-k} C(n-j-1, n-k-j).
Equals row reversal of triangle A112555 up to sign, where log(A112555) = A112555 - I. Unsigned row sums equals A052953 (Jacobsthal numbers + 1). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms. - Paul D. Hanna, Jan 20 2006
Sum_{k=0..n} T(n,k)*x^k = A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively (see the square array in A112739). - Philippe Deléham, Feb 22 2014

			Triangle starts
  1;
  1,  1;
  1,  0,  1;
  1,  1, -1,  1;
  1,  0,  2, -2,  1;
  1,  1, -2,  4, -3,  1;
  1,  0,  3, -6,  7, -4,  1;
Matrix log begins:
  0;
  1,  0;
  1,  0,  0;
  1,  1, -1,  0;
  1,  1,  1, -2,  0;
  1,  1,  1,  1, -3,  0; ...
Production matrix begins
  1,  1,
  0, -1,  1,
  0,  0, -1,  1,
  0,  0,  0, -1,  1,
  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0,  0, -1,  1.
- _Paul Barry_, Apr 08 2011

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
H. Belbachir and F. Bencherif, On some properties of bivariate Fibonacci and Lucas Polynomials, JIS 11 (2008) 08.2.6.
Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
H. Prodinger, On the expansion of Fibonacci and Lucas Polynomials, JIS 12 (2009) 09.1.6.

Cf. A174294, A174295, A174296, A174297. - Mats Granvik, Mar 15 2010
Cf. A072547 (central terms), A112555 (reversed rows), A112465, A052953, A112556, A112739, A119258.
See A279006 for another version.

T:= function(n,k)
    if k=0 or k=n then return 1;
    else return T(n-1,k-1) - T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 13 2019

a112468 n k = a112468_tabl !! n !! k
a112468_row n = a112468_tabl !! n
a112468_tabl = iterate (\xs -> zipWith (-) ([2] ++ xs) (xs ++ [0])) [1]
-- Reinhard Zumkeller, Jan 03 2014

function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1,k-1) - T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2019

T := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k,k+1)*hypergeom( [1,n+1],[k+2],m)/(k+1)!; A112468 := (n,k) -> T(n,n-k,-1);
seq(print(seq(simplify(A112468(n,k)),k=0..n)),n=0..10); # Peter Luschny, Jul 25 2014

T[n_, 0] = 1; T[n_, n_] = 1; T[n_, k_ ]:= T[n, k] = T[n-1, k-1] - T[n-1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Mar 06 2013 *)

{T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff((1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x),n,X),k,Y)} \\ Paul D. Hanna, Jan 20 2006

T(n,k) = if(k==0 || k==n, 1, T(n-1, k-1) - T(n-1, k)); \\ G. C. Greubel, Nov 13 2019

@CachedFunction
def T(n, k):
    if (k<0 or n<0): return 0
    elif (k==0 or k==n): return 1
    else: return T(n-1, k-1) - T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 13 2019

Triangle T(n,k) read by rows: T(n,0)=1, T(n,k) = T(n-1,k-1) - T(n-1,k). - Mats Granvik, Mar 15 2010
Number triangle T(n, k)= Sum_{j=0..n-k} C(n-j-1, n-k-j)*(-1)^(n-k-j).
G.f. of matrix power T^m: (1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x). G.f. of matrix log: x*(1-2*x*y+x^2*y)/(1-x*y)^2/(1-x). - Paul D. Hanna, Jan 20 2006
T(n, k) = R(n,n-k,-1) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k,k+1)*hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014

A014300 Number of nodes of odd outdegree in all ordered rooted (planar) trees with n edges.

Original entry on oeis.org

1, 2, 7, 24, 86, 314, 1163, 4352, 16414, 62292, 237590, 909960, 3497248, 13480826, 52097267, 201780224, 783051638, 3044061116, 11851853042, 46208337584, 180383564228, 704961896036, 2757926215742, 10799653176704, 42326626862636, 166021623024584, 651683311373788

Offset: 1

Emeric Deutsch

Also total number of blocks of odd size in all Catalan(n) possible noncrossing partitions of [n].
Convolution of the sequence of central binomial coefficients 1,2,6,20,70,... (A000984) and of the sequence of Fine numbers 1,0,1,2,6,18,... (A000957).
Row sums of A119307. - Paul Barry, May 13 2006
Hankel transform is A079935. - Paul Barry, Jul 17 2009
Also for n>=1 the number of unimodal functions f:[n]->[n] with f(i)<>f(i+1). a(3) = 7: [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,3,1], [2,3,2], [3,2,1]. - Alois P. Heinz, May 23 2013
Also, number of sets of n rational numbers on [0,1) such that if x belongs to the set, the fractional part of 2x also belongs to it. - Jianing Song and Andrew Howroyd, May 18 2018
Let A(i, j) denote the infinite array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i times to the function ((-1)^(n + 1) + 1)/2 for n > 0. Then A(n, n) equals a(n) for all n > 0. - John M. Campbell, Jan 20 2019
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p >= 3 and positive integers n and k. - Peter Bala, Jan 07 2022

Alois P. Heinz, Table of n, a(n) for n = 1..500
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Hacène Belbachir and Abdelghani Mehdaoui, Diagonal sums in Pascal pyramid (1, 2, r), Les Annales RECITS (2019) Vol. 6, 45-52.
N. Dershowitz and S. Zaks, Ordered trees and non-crossing partitions, Discrete Math., 62 (1986), 215-218.
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Index entries for sequences related to rooted trees

Cf. A059481, A000957, A000984, A072547, A119259, A119307, A079935, A333564.

[(&+[(-1)^(n-k)*Binomial(n+k-1, k-1): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Feb 19 2019

a:= proc(n) a(n):= `if`(n<3, n, ((12-40*n+21*n^2) *a(n-1)+
       2*(3*n-1)*(2*n-3) *a(n-2))/ (2*(3*n-4)*n))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 30 2012

Rest[CoefficientList[Series[2x/(1-4x+(1+2x)Sqrt[1-4x]),{x,0,40}],x]]  (* Harvey P. Dale, Apr 25 2011 *)
a[n_] := Sum[Binomial[2k, n-1], {k, 0, n-1}]; Array[a, 30] (* Jean-François Alcover, Dec 25 2015, after Paul Barry *)

a(n) = n--; sum(k=0, n, binomial(2*k,n)); \\ Michel Marcus, May 18 2018

from itertools import count, islice
def A014300_gen(): # generator of terms
    yield from (1,2)
    a, c = 1, 1
    for n in count(1):
        yield (a:=(3*n+5)*(c:=c*((n<<2)+2)//(n+2))-a>>1)
A014300_list = list(islice(A014300_gen(),20)) # Chai Wah Wu, Apr 26 2023

[sum((-1)^(n-k)*binomial(n+k-1, k-1) for k in (0..n)) for n in (1..30)] # G. C. Greubel, Feb 19 2019

a(n) = (binomial(2*n, n) + A000957(n))/3; [simplified by Alexander Burstein, Nov 24 2023]
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k-1). - Vladeta Jovovic, Aug 28 2002
G.f.: 2*z/(1-4*z+(1+2*z)*sqrt(1-4*z)).
a(n) = Sum_{j=0..floor((n-1)/2)} binomial(2*n-2*j-2, n-1).
2*a(n) + a(n-1) = (3*n-1)*Catalan(n-1). - Vladeta Jovovic, Dec 03 2004
a(n) = (-1)^n*Sum_{i=0..n} Sum_{j=n..2*n} (-1)^(i+j)*binomial(j, i). - Benoit Cloitre, Jun 18 2005
a(n) = Sum_{k=0..n} C(2*k,n) [offset 0]. - Paul Barry, May 13 2006
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n+k-1,k-1). - Paul Barry, Jul 18 2006
From Paul Barry, Jul 17 2009: (Start)
a(n) = Sum_{k=0..n} C(2*n-k,n-k)*(1+(-1)^k)/2.
a(n) = Sum_{k=0..n} C(n+k,k)*(1+(-1)^(n-k))/2. (End)
a(n) is the coefficient of x^(n+1)*y^(n+1) in 1/(1- x^2*y/((1-2*x)*(1-y))). - Ira M. Gessel, Oct 30 2012
a(n) = -binomial(2*n,n-1)*hyper2F1([1,2*n+1],[n+2], 2). - Peter Luschny, Jul 25 2014
a(n) = [x^n] x/((1 - x^2)*(1 - x)^n). - Ilya Gutkovskiy, Oct 25 2017
a(n) ~ 4^n / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 25 2017
D-finite with recurrence: 2*n*a(n) +(-3*n-4)*a(n-1) +2*(-9*n+19)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Feb 20 2020
a(n) = A333564(n)/2^n. - Peter Bala, Apr 09 2020
a(n) = (1/2)*(binomial(2*n,n) - A072547(n)). - Peter Bala, Mar 28 2023

A108561 Triangle read by rows: T(n,0)=1, T(n,n)=(-1)^n, T(n+1,k)=T(n,k-1)+T(n,k) for 0 < k < n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 2, 0, 1, 1, 3, 4, 2, 1, -1, 1, 4, 7, 6, 3, 0, 1, 1, 5, 11, 13, 9, 3, 1, -1, 1, 6, 16, 24, 22, 12, 4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, -1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, -1, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 0, 1, 1, 11, 56, 174, 367

Offset: 0

Reinhard Zumkeller, Jun 10 2005

Sum_{k=0..n} T(n,k) = A078008(n);
Sum_{k=0..n} abs(T(n,k)) = A052953(n-1) for n > 0;
T(n,1) = n - 2 for n > 1;
T(n,2) = A000124(n-3) for n > 2;
T(n,3) = A003600(n-4) for n > 4;
T(n,n-6) = A001753(n-6) for n > 6;
T(n,n-5) = A001752(n-5) for n > 5;
T(n,n-4) = A002623(n-4) for n > 4;
T(n,n-3) = A002620(n-1) for n > 3;
T(n,n-2) = A008619(n-2) for n > 2;
T(n,n-1) = n mod 2 for n > 0;
T(2*n,n) = A072547(n+1).
Sum_{k=0..n} T(n,k)*x^k = A232015(n), A078008(n), A000012(n), A040000(n), A001045(n+2), A140725(n+1) for x = 2, 1, 0, -1, -2, -3 respectively. - Philippe Deléham, Nov 17 2013, Nov 19 2013
(1,a^n) Pascal triangle with a = -1. - Philippe Deléham, Dec 27 2013
T(n,k) = A112465(n,n-k). - Reinhard Zumkeller, Jan 03 2014

			From _Philippe Deléham_, Nov 17 2013: (Start)
Triangle begins:
  1;
  1, -1;
  1,  0,  1;
  1,  1,  1, -1;
  1,  2,  2,  0,  1;
  1,  3,  4,  2,  1, -1;
  1,  4,  7,  6,  3,  0,  1; (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
C. Merino, S. D. Noble, M. Ramirez-Ibanez, R. Villarroel-Flores, On the structure of the h-vector of a paving matroid, Eur. J. Comb. 33, No. 8, 1787-1799 (2012).
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318 (a=1), A008949(a=2), A164844(a=10).
Similar to the triangles A035317, A059259, A080242, A112555.
Cf. A228196
Cf. A072547 (central terms).

Flat(List([0..13],n->List([0..n],k->Sum([0..k],i->Binomial(n,i)*(-2)^(k-i))))); # Muniru A Asiru, Feb 19 2018

a108561 n k = a108561_tabl !! n !! k
a108561_row n = a108561_tabl !! n
a108561_tabl = map reverse a112465_tabl
-- Reinhard Zumkeller, Jan 03 2014

A108561 := (n, k) -> add(binomial(n, i)*(-2)^(k-i), i = 0..k):
seq(seq(A108561(n,k), k = 0..n), n = 0..12); # Peter Bala, Feb 18 2018

Clear[t]; t[n_, 0] = 1; t[n_, n_] := t[n, n] = (-1)^Mod[n, 2]; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)

def A108561_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n, k) for k in (1..n-1)]+[(-1)^(n+1)]
for n in (1..12): print(A108561_row(n)) # Peter Luschny, Mar 16 2016

G.f.: (1-y*x)/(1-x-(y+y^2)*x). - Philippe Deléham, Nov 17 2013
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 17 2013
From Peter Bala, Feb 18 2018: (Start)
T(n,k) = Sum_{i = 0..k} binomial(n,i)*(-2)^(k-i), 0 <= k <= n.
The n-th row polynomial is the n-th degree Taylor polynomial of the rational function (1 + x)^n/(1 + 2*x) about 0. For example, for n = 4, (1 + x)^4/(1 + 2*x) = 1 + 2*x + 2*x^2 + x^4 + O(x^5). (End)

Definition corrected by Philippe Deléham, Dec 26 2013

p divides a((p+1)/2) for prime p = 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, ... (A033200: primes congruent to {1, 3} mod 8; or, odd primes of the form x^2 + 2*y^2).
p divides a((p-3)/2) for prime p = 17, 41, 73, 89, 97, 113, 137, ... (A007519: primes of the form 8n+1).
Essentially the same as partial sums of A072547. - Seiichi Manyama, Jan 30 2023

This sequence is the 4th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.

Apparently the number of peaks in all Dyck paths of semilength n+4 that are 2 steps higher than the preceding peak. - David Scambler, Apr 22 2013

Number of positive terms in expansion of (x_1+x_2+...+x_{n-1}-x_n)^(n+1). - Sergio Falcon, Feb 08 2007

Without the beginning "1" and "-1", we obtain the second diagonal over the principal diagonal of the array notified by B. Cloitre in A026641 and used by R. Choulet in A172025, and from A172061 to A172066. - Richard Choulet, Jan 25 2010

This sequence is the third diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows:
  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,
  1,  1,  2,  2,  3,  3,  4,  4,  5,  5,  6,  6,  7,  7,
  1,  2,  4,  6,  9, 12, 16, 20, 25, 30,
  1,  3,  7, 13, 22, 34, 50, 70, 95.
The Maple programs give the first diagonals of this array.
Apparently the number of peaks in all Dyck paths of semilength n+3 that are 1 step higher than the preceding peak. - David Scambler, Apr 22 2013

cp's OEIS Frontend

A000957 Fine's sequence (or Fine numbers): number of relations of valence >= 1 on an n-set; also number of ordered rooted trees with n nodes having root of even degree.

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A026641 Number of nodes of even outdegree (including leaves) in all ordered trees with n edges.

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A112555 Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I) and consequently the matrix logarithm satisfies log(T) = T - I, where I is the identity matrix.

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A112468 Riordan array (1/(1-x), x/(1+x)).

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A014300 Number of nodes of odd outdegree in all ordered rooted (planar) trees with n edges.

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A172061 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=4.

A172025 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=3.