A025566 -id:A025566 - cp's OEIS frontend

A001791 a(n) = binomial coefficient C(2n, n-1).

0, 1, 4, 15, 56, 210, 792, 3003, 11440, 43758, 167960, 646646, 2496144, 9657700, 37442160, 145422675, 565722720, 2203961430, 8597496600, 33578000610, 131282408400, 513791607420, 2012616400080, 7890371113950, 30957699535776, 121548660036300, 477551179875952

Offset: 0

N. J. A. Sloane

Number of peaks at even level in all Dyck paths of semilength n+1. Example: a(2)=4 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUUDDD, where U=(1,1), D=(1,-1) and the peaks at even level are shown by *. - Emeric Deutsch, Dec 05 2003
Also number of long ascents (i.e., ascents of length at least two) in all Dyck paths of semilength n+1. Example: a(2)=4 because in the five Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and (UUU)DDD, we have four long ascents (shown between parentheses). Here U=(1,1) and D=(1,-1). Also number of branch nodes (i.e., vertices of outdegree at least two) in all ordered trees with n+1 edges. - Emeric Deutsch, Feb 22 2004
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=1. Example: For n=2 these are the paths EENN, ENEN, ENNE and NEEN. - Herbert Kociemba, May 23 2004
Narayana transform (A001263) of [1, 3, 5, 7, 9, ...] = (1, 4, 15, 56, 210, ...). Row sums of triangles A136534 and A136536. - Gary W. Adamson, Jan 04 2008
Starting with offset 1 = the Catalan sequence starting (1, 2, 5, 14, ...) convolved with A000984: (1, 2, 6, 20, ...). - Gary W. Adamson, May 17 2009
Also number of peaks plus number of valleys in all Dyck n-paths. - David Scambler, Oct 08 2012
Apparently counts UDDUD in all Dyck paths of semilength n+2. - David Scambler, Apr 22 2013
Apparently the number of peaks strictly left of the midpoint in all Dyck paths of semilength n+1. - David Scambler, Apr 30 2013
For n>0, a(n) is the number of compositions of n into at most n parts if zeros are allowed as parts (so-called "weak" compositions). - L. Edson Jeffery, Jul 24 2014
Number of paths in the half-plane x >= 0, from (0,0) to (2n,2), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=2, we have the 4 paths: UUUD, UUDU, UDUU, DUUU. - José Luis Ramírez Ramírez, Apr 19 2015
For n>1, 1/a(n) is the probability that when a stick is broken up at n points independently and uniformly chosen at random along its length any triple of pieces of the n+1 pieces can form a triangle. The corresponding probability for the existence of at least one triple is A339392(n)/A339393(n). - Amiram Eldar, Dec 04 2020
a(n) is the number of lattice paths of 2n steps taken from the step set {U=(1,1), D=(1,-1)} that start at the origin, never go below the x-axis, and end strictly above the x-axis; more succinctly, proper left factors of Dyck paths. For example, a(2)=4 counts UUUU, UUUD, UUDU, UDUU. - David Callan and Emeric Deutsch, Jan 25 2021
From Gus Wiseman, Jul 21 2021: (Start)
Also the number of integer compositions of 2n+1 with alternating sum -1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(1) = 1 through a(3) = 15 compositions are:
  (1,2)  (2,3)      (3,4)
         (1,3,1)    (1,4,2)
         (1,1,1,2)  (2,4,1)
         (1,2,1,1)  (1,1,2,3)
                    (1,2,2,2)
                    (1,3,2,1)
                    (2,1,1,3)
                    (2,2,1,2)
                    (2,3,1,1)
                    (1,1,1,3,1)
                    (1,2,1,2,1)
                    (1,3,1,1,1)
                    (1,1,1,1,1,2)
                    (1,1,1,2,1,1)
                    (1,2,1,1,1,1)
The following relate to these compositions.
- The unordered version is A000070.
- Allowing any negative alternating sum gives A000346.
- The opposite (positive 1) version is A000984.
- The version for reverse-alternating sum is also A001791 (this sequence).
- Taking alternating sum -2 instead of -1 gives A002054.
- The shifted version for alternating sum 0 is counted by A088218 and ranked by A344619.
- Ranked by A345910 (reverse: A345912).
Equivalently, a(n) counts binary numbers with 2n+1 digits and one more 0 than 1's. For example, the a(2) = 4 binary numbers are: 10001, 10010, 10100, 11000.
(End)
The diagonal of a square n X n matrix where cells of the first row are the nonnegative integers and cells of subsequent rows are sums of cells of the previous row up to and including n. - Torlach Rush, Apr 24 2024
For n>=1, a(n) is the independence number of the odd graph O_{n+1}. - Miquel A. Fiol, Jul 07 2024

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Cornelius Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.
R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.
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).

T. D. Noe and Matuszka Tamás, Table of n, a(n) for n = 0..1200 (terms n = 0..200 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, Vol. 340, No. 10 (2017), pp. 2550-2558; preprint, 2016.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Norman Biggs, Some odd graph theory, Second International Conference on Combinatorial Mathematics, Annals of the New York Academy of Sciences 319 (1979), 71-81.
Miklós Bóna, Surprising Symmetries in Objects Counted by Catalan Numbers, Electronic J. Combin., 19 (2012), P62, eq. (6).
Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
Jelena Đokic, A short note on the order of the double reduced 2-factor transfer digraph for rectangular grid graphs, arXiv:2308.04155 [math.CO], 2023.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 21.
Guo-Niu Han, Enumeration of Standard Puzzles.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 260-288.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq., Vol. 17 (2014), Article 14.3.5.
Christian Krattenthaler and Daniel Yaqubi, Some determinants of path generating functions, II, Advances in Applied Mathematics, Vol. 101 (2018), pp. 232-265; arXiv preprint, arXiv:1802.05990 [math.CO], 2018.
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.3. - From _N. J. A. Sloane_, Sep 16 2012
Mark Shattuck, Enumeration of non-crossing partitions according to subwords with repeated letters, arXiv:2303.06300 [math.CO], 2023.
Zhi Lan Wang, Tautological integrals on symmetric products of curves, Acta Mathematica Sinica, English Series, Vol. 32, No. 8 (2016), pp. 901-910; arXiv preprint, arXiv:1506.08405 [math.AG], 2015-2016; alternative link.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Diagonal 3 of triangle A100257.
First differences are in A076540.
Cf. A000108, A000984, A002378, A002390, A025566, A132812.
A345197 counts compositions by length and alternating sum.
Cf. A000070, A000302, A000346, A002054, A008549, A032443, A088218, A097805, A163493, A202736, A345910.

List([0..30],n->Binomial(2*n,n-1)); # Muniru A Asiru, Aug 09 2018

[Binomial(2*n, n-1): n in [0..30]]; // Vincenzo Librandi, Apr 20 2015

Table[Binomial[2n,n-1],{n,0,30}] (* Harvey P. Dale, Jul 12 2012 *)
CoefficientList[ Series[(1 - 2x - Sqrt[1 - 4x])/(2x*Sqrt[1 - 4x]), {x, 0, 26}], x] (* Robert G. Wilson v, Aug 10 2018 *)

A001791(n):=binomial(2*n,n-1)$
makelist(A001791(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
a(n)=if(n<1,0,(2*n)!/(n+1)!/(n-1)!)
```

a(n) = n*A000108(n).
G.f.: x*(d/dx)c(x) where c(x) = Catalan g.f. - Wolfdieter Lang
Convolution of A001700 (central binomial of odd order) and A000108 (Catalan): a(n+1) = Sum_{k=0..n} C(k)*binomial(2*(n-k)+1, n-k), C(k): Catalan. - Wolfdieter Lang
E.g.f.: exp(2x) * I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 08 2002
a(n) = Sum_{k=0..n} C(n, k)*C(n, k+1). - Paul Barry, May 15 2003
a(n) = Sum_{i=1..n} binomial(i+n-1, n).
G.f.: (1-2x-sqrt(1-4x))/(2x*sqrt(1-4x)). - Emeric Deutsch, Dec 05 2003
a(n) = A092956/(n!). - Amarnath Murthy, Jun 16 2004
a(n) = binomial(2n,n) - A000108(n). - Paul Barry, Apr 21 2005
a(n) = (1/(2*Pi))*Integral_{x=0..4} (x^n*(x-2)/sqrt(x(4-x))) is the moment sequence representation. - Paul Barry, Jan 11 2007
Row sums of triangle A132812 starting (1, 4, 15, 56, 210, ...). - Gary W. Adamson, Sep 01 2007
Starting (1, 4, 15, 56, 210, ...) gives the binomial transform of A025566 starting (1, 3, 8, 22, 61, 171, ...). - Gary W. Adamson, Sep 01 2007
For n >= 1, a(2^n) = 2^(n+1)*A001795(2^(n-1)). - Vladimir Shevelev, Sep 05 2010
D-finite with recurrence: (n-1)*(n+1)*a(n) = 2*n*(2n-1)*a(n-1). - R. J. Mathar, Dec 17 2011
From Sergei N. Gladkovskii, Jul 07 2012: (Start)
G.f.: -1/(2*x) - G(0) where G(k) = 1 - 1/(2*x - 8*x^3*(2*k+1)/(4*x^2*(2*k+1)- (k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step);
E.g.f.: BesselI(1,2*x)*exp(2*x) = x*G(0) where G(k) = 1 + 2*x*(4*k+3)/((2*k+1)*(2*k+3) - x*(2*k+1)*(2*k+3)*(4*k+5)/(x*(4*k+5) + 2*(k+1)*(k+2)/G(k+1))); (continued fraction, 3rd kind, 3-step).
(End)
G.f.: c(x)^3/(2-c(x)) where c(x) is the g.f. for A000108. - Cheyne Homberger, May 05 2014
G.f.: z*C(z)^2/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015
G.f.: x*2F1(3/2,2;3;4x). - R. J. Mathar, Aug 09 2015
a(n) = Sum_{i=1..n} binomial(2*i-2,i-1)*binomial(2*(n-i+1),n-i+2)/(n-i+1). - Vladimir Kruchinin, Sep 07 2015
L.g.f.: 1/(1 - x/(1 - x/(1 - x/(1 - x/(1 - x/(1 - ...)))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017
Sum_{n>=1} 1/a(n) = 1/3 + 5*Pi/(9*sqrt(3)). - Amiram Eldar, Dec 04 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/5 + 14*sqrt(5)*log(phi)/25, where log(phi) = A002390. - Amiram Eldar, Feb 20 2021
a(n) = Product_{i=1..(n - 1)} (((4*i + 6)*i + 2)/((i + 2)*i)), for n>=1. - Neven Sajko, Oct 10 2021
a(n) = 2^(2*n)*gamma(n + 1/2)/(sqrt(Pi)*gamma(n)*(n+1)) for n > 0, and a(0) = lim_{n->0} a(n). - Karol A. Penson, Apr 24 2025

A025564 Triangular array, read by rows: pairwise sums of trinomial array A027907.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 4, 3, 1, 1, 4, 8, 10, 8, 4, 1, 1, 5, 13, 22, 26, 22, 13, 5, 1, 1, 6, 19, 40, 61, 70, 61, 40, 19, 6, 1, 1, 7, 26, 65, 120, 171, 192, 171, 120, 65, 26, 7, 1, 1, 8, 34, 98, 211, 356, 483, 534, 483, 356, 211, 98, 34, 8, 1, 1, 9, 43, 140, 343, 665, 1050, 1373

Offset: 0

Clark Kimberling

Counting the top row as row 0, T(n,k) is the number of strings of nonnegative integers "s(1)s(2)s(3)...s(k)" such that s(1)+s(2)+s(3)+...+s(k) = n and the string does not contain the substring "00". E.g., T(3,5) = 8 because the valid strings are 02010, 01020, 11010, 10110, 10101, 01110, 01101 and 01011. T(4,3) = 13, counting 040, 311, 301, 130, 031, 103, 013, 220, 202, 022, 211, 121 and 112. - Jose Luis Arregui (arregui(AT)unizar.es), Dec 05 2007

			                  1
              1   2   1
          1   3   4   3   1
      1   4   8  10   8   4   1
  1   5  13  22  26  22  13   5   1

Columns include A025565, A025566, A025567, A025568.

Cf. A025177.

T[, 0] = 1; T[1, 1] = 2; T[n, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Jul 22 2018 *)

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,2,1][k+1], if( n==2, [1,3,4,3,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};

T(n,k)=polcoeff(Ser(polcoeff(Ser((1+y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)

{T(n, k) = if( k<0 || k>2*n, 0, if(n==0, 1, polcoeff( (1 + x + x^2)^n, k)+ polcoeff( (1 + x + x^2)^(n-1), k-1)))};

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 2, 1], [1, 3, 4, 3, 1].

G.f.: (1+yz)/[1-z(1+y+y^2)].

Edited by Ralf Stephan, Jan 09 2005

Edited by Clark Kimberling, Jun 20 2012

A026105 Triangle T read by rows: differences of Motzkin triangle (A026300).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 3, 6, 7, 5, 1, 4, 10, 16, 18, 12, 1, 5, 15, 30, 44, 46, 30, 1, 6, 21, 50, 89, 120, 120, 76, 1, 7, 28, 77, 160, 259, 329, 316, 196, 1, 8, 36, 112, 265, 496, 748, 904, 841, 512, 1, 9, 45, 156, 413, 873, 1509, 2148, 2493, 2257, 1353, 1, 10, 55, 210, 614, 1442, 2795, 4530, 6150, 6898, 6103, 3610

Offset: 0

Clark Kimberling

For n >= 2, T(n,k)= number of nonnegative integer strings s(0),...,s(n) such that s(n)=n-k, s(0)=s(1)=1, |s(i)-s(i-1)|<=1 for i >= 2.

			1
1,1
1,1,1
1,2,3,2
1,3,6,7,5
1,4,10,16,18,12
1,5,15,30,44,46,30

Right-hand columns include A002026, A026107, A026134, A026109, A026110.

Row sums are in A025566. Central column is in A026112.

T(n, k) = A026300(n, k) - A026300(n-1, k-1), T(1, 1) = 1.
T(i, 0)=1 for i >= 0, T(2, 1)=1, T(2, 2)=1, T(3, 1)=2, T(3, 2)=3, T(3, 3)=2; and for i >= 4, T(i, 1)=i-1, T(i, i)=T(i-1, i-2)+T(i-1, i-1) and T(i, j)=T(i-1, j-2)+T(i-1, j-1)+T(i-1, j) for j=2, 3, ...., i-1.
Right-hand columns have g.f. (1-z)*M^k, where M is g.f. of Motzkin numbers (A001006).

Edited by Ralf Stephan, Dec 18 2004
a(65) corrected and more terms from Sean A. Irvine, Sep 16 2019
Offset set to 0 by Alois P. Heinz, Sep 16 2019

A026135 Number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T defined in A026120.

Original entry on oeis.org

1, 2, 5, 14, 39, 110, 312, 890, 2550, 7334, 21161, 61226, 177575, 516114, 1502867, 4383462, 12804429, 37452870, 109682319, 321563658, 943701141, 2772060618, 8149661730, 23978203662, 70600640796, 208014215066, 613266903927

Offset: 0

Clark Kimberling

nonn

a(n) is the total number of rows of consecutive peaks in all Motzkin (n+2)-paths. For example, with U=upstep, D=downstep, F=flatstep, the path FU(UD)FU(UDUDUD)DD(UD) contains 3 rows of peaks (in parentheses). The 9 Motzkin 4-paths are FFFF, FF(UD), F(UD)F, FUFD, (UD)FF, (UDUD), UFDF, UFFD, U(UD)D, containing a total of 5 rows of peaks and so a(2)=5. - David Callan, Aug 16 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000

First differences are in A025566, second differences in A005773.

Pairwise sums of A025179.

CoefficientList[Series[((x - 1)^2*((1 + x)/(1 - 3 x))^(1/2) + x^2 - 1)/(2*x^2), {x,0,50}], x] (* G. C. Greubel, May 22 2017 *)

x='x+O('x^50); Vec(((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2)) \\ G. C. Greubel, May 22 2017

a(n) = Sum_{k=0..n} binomial(n-1, k-1)*binomial(k+1, floor((k+1)/2)). - Vladeta Jovovic, Sep 18 2003
G.f.: ((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2). - David Callan, Aug 16 2006
G.f. = (1+z)*(1+z^2)/(1-z) where z=x*A001006(x). [From R. J. Mathar, Jul 07 2009]
Conjecture: (n+2)*a(n) +3*(-n-1)*a(n-1) +(-n-2)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 23 2013

More terms from David Callan, Aug 16 2006

Typo in a(19) corrected by R. J. Mathar, Jul 07 2009

A115990 Riordan array (1/sqrt(1-2x-3x^2), (1-2x-3x^2)/(2(1-3x)) - sqrt(1-2x-3x^2)/2).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 5, 3, 1, 19, 13, 8, 4, 1, 51, 35, 22, 12, 5, 1, 141, 96, 61, 35, 17, 6, 1, 393, 267, 171, 101, 53, 23, 7, 1, 1107, 750, 483, 291, 160, 77, 30, 8, 1, 3139, 2123, 1373, 839, 476, 244, 108, 38, 9, 1, 8953, 6046, 3923, 2423, 1406, 752, 360, 147, 47, 10

Offset: 0

Paul Barry, Feb 10 2006

First column is central trinomial coefficients A002426. Second column is number of directed animals of size n+1, A005773(n+1). Row sums are A005717 (number of horizontal steps in all Motzkin paths of length n). First column has e.g.f. exp(x) I_0(2x). Row sums have e.g.f. dif(exp(x) I_1(2x),x).

Riordan array (1/sqrt(1-2*x-3*x^2), (1+x-sqrt(1-2*x-3*x^2))/2).

			Triangle begins
    1;
    1,  1;
    3,  2,  1;
    7,  5,  3,  1;
   19, 13,  8,  4,  1;
   51, 35, 22, 12,  5,  1;
  141, 96, 61, 35, 17,  6,  1;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A115991, A005773 (k=1), A025566 (k=2), A035045 (k=3), A152948 (diag. n=k+2), .

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)) ))); # G. C. Greubel, May 09 2019

[[(&+[Binomial(n-k, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019

A115990 := proc(n,k)
    add(binomial(n-k,j-k)*binomial(j,n-j),j=0..n) ;
end proc:
seq(seq(A115990(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 25 2023

Table[Sum[ Binomial[n-k, j-k]*Binomial[j, n-j], {j, 0, n}], {n, 0, 10}, {k, 0, n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

{T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j))}; \\ G. C. Greubel, May 09 2019

[[sum(binomial(n-k, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019

Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j).

cp's OEIS Frontend

A001791 a(n) = binomial coefficient C(2n, n-1).

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A025564 Triangular array, read by rows: pairwise sums of trinomial array A027907.

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A026105 Triangle T read by rows: differences of Motzkin triangle (A026300).

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A026135 Number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T defined in A026120.

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A115990 Riordan array (1/sqrt(1-2*x-3*x^2), (1-2*x-3*x^2)/(2*(1-3*x)) - sqrt(1-2*x-3*x^2)/2).

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A378816 Expansion of 2*(x - 1)^3/(3*x^3 - 5*x^2 + x + 1 + sqrt(-(x - 1)^3*(x + 1)^2*(3*x + 1))).

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PARI

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A102003 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k branches of odd length (n>=0, 0<=k<=n).

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A115990 Riordan array (1/sqrt(1-2x-3x^2), (1-2x-3x^2)/(2(1-3x)) - sqrt(1-2x-3x^2)/2).

A378816 Expansion of 2(x - 1)^3/(3x^3 - 5x^2 + x + 1 + sqrt(-(x - 1)^3(x + 1)^2(3x + 1))).