A014318 -id:A014318 - cp's OEIS frontend

A000346 a(n) = 2^(2n+1) - binomial(2n+1, n+1).

1, 5, 22, 93, 386, 1586, 6476, 26333, 106762, 431910, 1744436, 7036530, 28354132, 114159428, 459312152, 1846943453, 7423131482, 29822170718, 119766321572, 480832549478, 1929894318332, 7744043540348, 31067656725032, 124613686513778, 499744650202436

Offset: 0

N. J. A. Sloane, Wolfdieter Lang

Also a(n) = 2nd elementary symmetric function of binomial(n,0), binomial(n,1), ..., binomial(n,n).
Also a(n) = one half the sum of the heights, over all Dyck (n+2)-paths, of the vertices that are at even height and terminate an upstep. For example with n=1, these vertices are indicated by asterisks in the 5 Dyck 3-paths: UU*UDDD, UU*DU*DD, UDUU*DD, UDUDUD, UU*DDUD, yielding a(1)=(2+4+2+0+2)/2=5. - David Callan, Jul 14 2006
Hankel transform is (-1)^n*(2n+1); the Hankel transform of sum(k=0..n, C(2*n,k) ) - C(2n,n) is (-1)^n*n. - Paul Barry, Jan 21 2007
Row sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/2) (A187926). - Emanuele Munarini, Mar 16 2011
From Gus Wiseman, Jul 19 2021: (Start)
For n > 0, a(n-1) is also the number of integer compositions of 2n with nonzero alternating sum, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A053754 /\ A345921. For example, the a(3-1) = 22 compositions of 6 are:
  (6)  (1,5)  (1,1,4)  (1,1,1,3)  (1,1,1,1,2)
       (2,4)  (1,2,3)  (1,1,3,1)  (1,1,2,1,1)
       (4,2)  (1,4,1)  (1,2,1,2)  (2,1,1,1,1)
       (5,1)  (2,1,3)  (1,3,1,1)
              (2,2,2)  (2,1,2,1)
              (3,1,2)  (3,1,1,1)
              (3,2,1)
              (4,1,1)
(End)

			G.f. = 1 + 5*x + 22*x^2 + 93*x^3 + 386*x^4 + 1586*x^5 + 6476*x^6 + ...

T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.
D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..500
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004.
Vijay Balasubramanian, Javier M. Magan, and Qingyue Wu, A Tale of Two Hungarians: Tridiagonalizing Random Matrices, arXiv:2208.08452 [hep-th], 2022.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271; see p. 270.
D. E. Davenport, L. K. Pudwell, L. W. Shapiro and L. C. Woodson, The Boundary of Ordered Trees, 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 185
R. K. Guy, Letter to N. J. A. Sloane
Toufik Mansour and José L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7. - From _N. J. A. Sloane_, Nov 25 2012
D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (A_n for s=2).
Vera Posch, Correlators in Matrix Models, Master Thesis, Uppsala Univ. (Sweden 2023). See p. 44.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
W. T. Tutte, On the enumeration of planar maps. Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218 (eq. 5, b=1).
N. J. A. Sloane, Notes

Cf. A000108, A014137, A014318. A column of A058893. Subdiagonal of A053979.
Bisection of A058622 and (possibly) A007008.
Cf. A045621, A068551.
Cf. A033504, A038806, A055217, A069731.
Even bisection of A294175 (without the first two terms).
The following relate to compositions of 2n with alternating sum k.
- The k > 0 case is counted by A000302.
- The k <= 0 case is counted by A000302.
- The k != 0 case is counted by A000346 (this sequence).
- The k = 0 case is counted by A001700 or A088218.
- The k < 0 case is counted by A008549.
- The k >= 0 case is counted by A114121.
A011782 counts compositions.
A086543 counts partitions with nonzero alternating sum (bisection: A182616).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A345197 counts compositions by length and alternating sum.
Cf. A000070, A001791, A007318, A025047, A027306, A032443, A053754, A120452, A163493, A239830, A344611, A345921.

[2^(2*n+1) - Binomial(2*n+1,n+1): n in [0..30]]; // Vincenzo Librandi, Jun 07 2011

seq(2^(2*n+1)-binomial(2*n,n)*(2*n+1)/(n+1), n=0..12); # Emanuele Munarini, Mar 16 2011

Table[2^(2n+1)-Binomial[2n,n](2n+1)/(n+1),{n,0,20}] (* Emanuele Munarini, Mar 16 2011 *)
a[ n_] := If[ n<-4, 0, (4^(n + 1) - Binomial[2 n + 2, n + 1]) / 2]; (* Michael Somos, Jan 25 2014 *)

makelist(2^(2*n+1)-binomial(2*n,n)*(2*n+1)/(n+1),n,0,12); /* Emanuele Munarini, Mar 16 2011 */

{a(n) = if( n<-4, 0, n++; (2^(2*n) - binomial(2*n, n)) / 2)}; /* Michael Somos, Jan 25 2014 */

G.f.: c(x)/(1-4x), c(x) = g.f. of Catalan numbers.
Convolution of Catalan numbers and powers of 4.
Also one half of convolution of central binomial coeffs. A000984(n), n=0, 1, 2, ... with shifted central binomial coeffs. A000984(n), n=1, 2, 3, ...
a(n) = A045621(2n+1) = (1/2)*A068551(n+1).
a(n) = Sum_{k=0..n} A000984(k)*A001700(n-k). - Philippe Deléham, Jan 22 2004
a(n) = Sum_{k=0..n+1} binomial(n+k, k-1)2^(n-k+1). - Paul Barry, Nov 13 2004
a(n) = Sum_{i=0..n} binomial(2n+2, i). See A008949. - Ed Catmur (ed(AT)catmur.co.uk), Dec 09 2006
a(n) = Sum_{k=0..n+1, C(2n+2,k)} - C(2n+2,n+1). - Paul Barry, Jan 21 2007
Logarithm g.f. log(1/(2-C(x)))=sum(n>0, a(n)/n*x^n), C(x)=(1-sqrt(1-4*x))/2x (A000108). - Vladimir Kruchinin, Aug 10 2010
D-finite with recurrence: (n+3) a(n+2) - 2(4n+9) a(n+1) + 8(2n+3) a(n) = 0. - Emanuele Munarini, Mar 16 2011
E.g.f.:exp(2*x)*(2*exp(2*x) - BesselI(0,2*x) - BesselI(1,2*x)).
This is the first derivative of exp(2*x)*(exp(2*x) - BesselI(0,2*x))/2. See the e.g.f. of A032443 (which has a plus sign) and the remarks given there. - Wolfdieter Lang, Jan 16 2012
a(n) = Sum_{0<=iMircea Merca, Apr 05 2012
A000346 = A004171 - A001700. See A032443 for the sum. - M. F. Hasler, Jan 02 2014
0 = a(n) * (256*a(n+1) - 224*a(n+2) + 40*a(n+3)) + a(n+1) * (-32*a(n+1) + 56*a(n+2) - 14*a(n+3)) + a(n+2) * (-2*a(n+2) + a(n+3)) if n>-5. - Michael Somos, Jan 25 2014
REVERT transform is [1,-5,28,-168,1056,...] = alternating signed version of A069731. - Michael Somos, Jan 25 2014
Convolution square is A038806. - Michael Somos, Jan 25 2014
BINOMIAL transform of A055217(n-1) is a(n-1). - Michael Somos, Jan 25 2014
(n+1) * a(n) = A033504(n). - Michael Somos, Jan 25 2014
Recurrence: (n+1)*a(n) = 512*(2*n-7)*a(n-5) + 256*(13-5*n)*a(n-4) + 64*(10*n-17)*a(n-3) + 32*(4-5*n)*a(n-2) + 2*(10*n+1)*a(n-1), n>=5. - Fung Lam, Mar 21 2014
Asymptotic approximation: a(n) ~ 2^(2n+1)*(1-1/sqrt(n*Pi)). - Fung Lam, Mar 21 2014
a(n) = Sum_{m = n+2..2*(n+1)} binomial(2*(n+1), m), n >= 0. - Wolfdieter Lang, May 22 2015
a(n) = A000302(n) + A008549(n). - Gus Wiseman, Jul 19 2021
a(n) = Sum_{j=1..n+1} Sum_{k=1..j} 2^(j-k)*binomial(n+k-1, n). - Fabio Visonà, May 04 2022
a(n) = (1/2)*(-1)^n*binomial(-(n+1), n+2)*hypergeom([1, 2*n + 3], [n + 3], 1/2). - Peter Luschny, Nov 29 2023

Corrected by Christian G. Bower

A064613 Second binomial transform of the Catalan numbers.

Original entry on oeis.org

1, 3, 10, 37, 150, 654, 3012, 14445, 71398, 361114, 1859628, 9716194, 51373180, 274352316, 1477635912, 8016865533, 43773564294, 240356635170, 1326359740956, 7351846397334, 40913414754324, 228508350629892

Offset: 0

Karol A. Penson, Sep 24 2001

nonn

Exponential convolution of Catalan numbers and powers of 2. - Vladeta Jovovic, Dec 03 2004
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and those at a higher level come in 4 colors. Example: a(3)=37 because, denoting  U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 4 paths of shape UHD. - Emeric Deutsch, May 02 2011
a(n) is the number of Schroeder paths of semilength n in which the (2,0)-steps come in 2 colors and having no (2,0)-steps at levels 1,3,5,... - José Luis Ramírez Ramírez, Mar 30 2013
From Tom Copeland, Nov 08 2014: (Start)
This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t)=x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf A126930.)
O.g.f.: G(x) = C[P[P(x,-1),-1]] = C[P(x,-2)] = (1-sqrt(1-4*x/(1-2*x)))/2 = x*A064613(x).
Ginv(x) =  Pinv[Cinv(x),-2] = P[Cinv(x),2] = x(1-x)/[1+2x(1-x)] = (x-x^2)/[1+2(x-x^2)] = x - 3 x^2 + 8 x^3 - ... is -A155020(-x) ignoring first term there. (Cf. A146559, A125145.)(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv:1110.6638 [math.NT], 2011.
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.

Cf. A007317, A000108, A014318.

Cf. A125145, A146559, A126930.

I:=[3,10]; [1] cat [n le 2 select I[n] else ((8*n-2)*Self(n-1)-(12*n-12)*Self(n-2))div (n+1): n in [1..30]]; // Vincenzo Librandi, Jan 23 2017

CoefficientList[Series[(1-Sqrt[(1-6*x)/(1-2*x)])/2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 29 2013 *)
a[n_] := 2^n Hypergeometric2F1[1/2, -n, 2, -2];
Array[a, 22, 0] (* Peter Luschny, Jan 27 2020 *)

x='x+O('x^66); Vec((1-sqrt((1-6*x)/(1-2*x)))/(2*x)) /* Joerg Arndt, Mar 31 2013 */

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)*2^(n-k)/(k+1).
a(n) = 2^n*hypergeom([1/2, -n], [2], -2).
G.f.: (1-sqrt((1-6*x)/(1-2*x)))/(2*x). - Vladeta Jovovic, May 03 2003
With offset 1: a(1) = 1, a(n) = 2^(n-1) + Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
D-finite with recurrence (n+1)*a(n) = (8*n-2)*a(n-1) - (12*n-12)*a(n-2). - Vladeta Jovovic, Jul 16 2004
E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x)). - Vladeta Jovovic, Dec 03 2004
Inverse binomial transform of A104455. - Philippe Deléham, Nov 30 2007
G.f.: 1/(1-3*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-... (continued fraction). - Paul Barry, Jul 02 2009
a(n) = Sum_{0<=k<=n} A052179(n,k)*(-1)^k. - Philippe Deléham, Nov 28 2009
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:
  3, 1, 0, 0, ...
  1, 3, 1, 0, ...
  1, 1, 3, 1, ...
  1, 1, 1, 3, ...
  ... (End)
a(n) ~ 2^(n-3/2)*3^(n+3/2)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jun 29 2013
G.f. A(x) satisfies: A(x) = 1/(1 - 2*x) + x * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020

Name clarified using a comment of Vladeta Jovovic by Peter Bala, Jan 27 2020

A097331 Expansion of 1 + 2x/(1 + sqrt(1 - 4x^2)).

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0

Offset: 0

Paul Barry, Aug 05 2004

Binomial transform is A097332. Second binomial transform is A014318.
Essentially the same as A126120. - R. J. Mathar, Jun 15 2008
Hankel transform is A087960(n) = (-1)^binomial(n+1,2). - Paul Barry, Aug 10 2009

Michael De Vlieger, Table of n, a(n) for n = 0..3340
Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, Motzkin paths with a restricted first return decomposition, Integers (2019) Vol. 19, A46.

A097331_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]-(-1)^w*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a,list)end: A097331_list(48); # Peter Luschny, May 19 2011

a[0] = 1; a[n_?OddQ] := CatalanNumber[(n-1)/2]; a[] = 0; Table[a[n], {n, 0, 48}] (* _Jean-François Alcover, Jul 24 2013 *)

def A097331_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 1; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] -= D[k-1]
            h += 1; R.append(abs(D[1]))
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
    return R
A097331_list(49) # Peter Luschny, Jun 03 2012

a(n) = 0^n + Catalan((n-1)/2)(1-(-1)^n)/2.
Unsigned version of A090192, A105523. - Philippe Deléham, Sep 29 2006
From Paul Barry, Aug 10 2009: (Start)
G.f.: 1+xc(x^2), c(x) the g.f. of A000108;
G.f.: 1/(1-x/(1+x/(1+x/(1-x/(1-x/(1+x/(1+x/(1-x/(1-x/(1+... (continued fraction);
G.f.: 1+x/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). (End)
G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+2*x) (continued fraction); more generally g.f. C(x/(1+2*x)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011
Conjecture: (n+1)*a(n) + n*a(n-1) + 4*(-n+2)*a(n-2) + 4*(-n+3)*a(n-3)=0. - R. J. Mathar, Dec 02 2012
Recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=a(1)=1. For nonzero terms, a(n) ~ 2^(n+1)/((n+1)^(3/2)*sqrt(2*Pi)). - Fung Lam, Mar 17 2014

A106270 Inverse of number triangle A106268; triangle T(n,k), 0 <= k <= n.

Original entry on oeis.org

1, -1, 1, -2, -1, 1, -5, -2, -1, 1, -14, -5, -2, -1, 1, -42, -14, -5, -2, -1, 1, -132, -42, -14, -5, -2, -1, 1, -429, -132, -42, -14, -5, -2, -1, 1, -1430, -429, -132, -42, -14, -5, -2, -1, 1, -4862, -1430, -429, -132, -42, -14, -5, -2, -1, 1, -16796, -4862, -1430, -429, -132, -42, -14, -5, -2, -1, 1

Offset: 0

Paul Barry, Apr 28 2005

Sequence array for the sequence a(n) = 2*0^n - C(n), where C = A000108 (Catalan numbers). Row sums are A106271. Antidiagonal sums are A106272.

The lower triangular matrix |T| (unsigned case) gives the Riordan matrix R = (c(x), x), a Toeplitz matrix. It is its own so called L-Eigen-matrix (cf. Bernstein - Sloane for such Eigen-sequences, and Barry for such eigentriangles), that is R*R = L*(R - I), with the infinite matrices I (identity) and L with matrix elements L(i, j) = delta(i,j-1) (Kronecker symbol; first upper diagonal with 1s). Thus R = L*(I - R^{-1}), and R^{-1} = I - L^{tr}*R (tr for transposed) is the Riordan matrix (1 - x*c(x), x) given in A343233. (For finite N X N matrices the R^{-1} equation is also valid, but for the other two ones the last row with only zeros has to be omitted.) - Gary W. Adamson and Wolfdieter Lang, Apr 11 2021

			Triangle (with rows n >= 0 and columns k >= 0) begins as follows:
     1;
    -1,    1;
    -2,   -1,   1;
    -5,   -2,  -1,   1;
   -14,   -5,  -2,  -1,  1;
   -42,  -14,  -5,  -2, -1,  1;
  -132,  -42, -14,  -5, -2, -1,  1;
  -429, -132, -42, -14, -5, -2, -1,  1;

Michel Marcus, Table of n, a(n) for n = 0..1325 (Rows n = 0..50 of triangle, flattened).
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

Cf. A000108, A014318, A106268, A106271, A106272, A112696, A343233.

A106270:= func< n,k | k eq n select 1 else -Catalan(n-k) >;
[A106270(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 09 2023

A106270[n_, k_]:= If[k==n, 1, -CatalanNumber[n-k]];
Table[A106270[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 09 2023 *)

C(n) = binomial(2*n,n)/(n+1); \\ A000108
T(n, k) = if(k <= n, 2*0^(n-k) - C(n-k), 0); \\ Michel Marcus, Nov 11 2022

def A106270(n,k): return 1 if (k==n) else -catalan_number(n-k)
flatten([[A106270(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 09 2023

Number triangle T(n, k) = 2*0^(n-k) - C(n-k) if k <= n, 0 otherwise; Riordan array (2*sqrt(1-4*x)/(1+sqrt(1-4*x)), x) = (c(x)*sqrt(1-4*x), x), where c(x) is the g.f. of A000108.
Sum_{k=0..n} T(n, k) = A106271(n).
Sum_{k=0..floor(n/2)} T(n, k) = A106272(n).
Bivariate g.f.: Sum_{n, k >= 0} T(n,k)*x^n*y^k = (1/(1 - x*y)) * (2 - c(x)), where c(x) is the g.f. of A000108. - Petros Hadjicostas, Jul 15 2019
From G. C. Greubel, Jan 09 2023: (Start)
Sum_{k=0..n} 2^(n-j)*abs(T(n,k)) = A112696(n).
Sum_{k=0..n} 2^k*abs(T(n,k)) = A014318(n). (End)

A097332 Expansion of (1/(1-x))(1+2x/(1-x+sqrt(1-2x-3x^2))).

Original entry on oeis.org

1, 2, 3, 5, 9, 18, 39, 90, 217, 540, 1375, 3563, 9361, 24872, 66707, 180341, 490913, 1344380, 3701159, 10237541, 28436825, 79288844, 221836403, 622599626, 1752360041, 4945087838, 13988490339, 39658308815, 112666081617

Offset: 0

Paul Barry, Aug 05 2004

Binomial transform of A097331. Binomial transform is A014318. Partial sums of 1+2x/(1-x+sqrt(1-2x-3x^2)) or (1+x+sqrt(1-2x-3x^2))/(1-x+sqrt(1-2x-3x^2)), which is A001006 with an extra leading 1.
Apparently the Motzkin transform of 1, 2, bar(1, -1, -1, 1), where bar() denotes a periodically continued series, as in A057077. - R. J. Mathar, Dec 11 2008
Starting with offset 1 = iterates of M * [1,1,0,0,0,...] where M = a tridiagonal matrix with [1,1,1,...] in the main and superdiagonals and [0,1,1,1,...] in the subdiagonal. - Gary W. Adamson, Jan 08 2009
Hankel transform is A087960(n) = (-1)^binomial(n+1,2). - Paul Barry, Aug 10 2009

			G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 9*x^4 + 18*x^5 + 39*x^6 + 90*x^7 + 217*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215. [See S_n on page 7.]

CoefficientList[Series[1/(1-x)*(1+(2x)/(1-x+Sqrt[1-2x-3x^2])),{x,0,40}],x] (* Harvey P. Dale, May 03 2012 *)
a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 2 x - 3 x^2]) / (2 x (1 - x)), {x, 0, n}]; (* Michael Somos, May 19 2014 *)

{a(n) = if( n<0, 0, polcoeff( (1 + x - sqrt(1 - 2*x - 3*x^2 + x^2 * O(x^n))) / (2 * x * (1 - x)), n))}; /* Michael Somos, May 19 2014 */

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(n, k)*Sum_{i=0..k} Catalan(k-i)*2^i.
G.f.: 1/(1-x-x/(1+x/(1-x+x/(1-x/(1-x-x/(1+x/(1-x+x/(1-x/(1-x-x/(1+... (continued fraction). - Paul Barry, Aug 10 2009
Conjecture D-finite with recurrence: (n+1)*a(n) - 3*n*a(n-1) + (-n+5)*a(n-2) + 3*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
a(n) ~ 3^(n+3/2) / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014
0 = a(n)*(9*a(n+1) + 6*a(n+2) - 27*a(n+3) + 12*a(n+4)) + a(n+1)*(-12*a(n+1) + 10*a(n+2) + 12*a(n+3) - 7*a(n+4)) + a(n+2)*(-12*a(n+2) + 14*a(n+3) - 6*a(n+4)) + a(n+3)*(a(n+4)). - Michael Somos, May 19 2014

A158495 Expansion of ((1-4x) + sqrt(1-4x))/(2(1-2x)).

Original entry on oeis.org

1, -1, -3, -8, -21, -56, -154, -440, -1309, -4048, -12958, -42712, -144210, -496432, -1735764, -6145968, -21986781, -79331232, -288307254, -1054253208, -3875769606, -14315659632, -53097586284, -197677736208, -738415086066

Offset: 0

Paul Barry, Mar 20 2009

Hankel transform is A158496.

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

Essentially the same as A014318, up to sign and offset.

Cf. A126966, A158496, A246432.

A158495:= func< n | n eq 0 select 1 else - (&+[2^(n-j)*Catalan(j-1): j in [1..n]]) >;
[A158495(n): n in [0..40]]; // G. C. Greubel, Jan 09 2023

CoefficientList[Series[((1-4x)+Sqrt[1-4x])/(2(1-2x)),{x,0,30}],x] (* Harvey P. Dale, Dec 15 2011 *)

def A158495(n): return int(n==0) - sum(2^(n-k)*catalan_number(k-1) for k in range(1,n+1))
[A158495(n) for n in range(41)] # G. C. Greubel, Jan 09 2023

a(n) = (2*0^n - 2^n + A126966(n))/2.
Conjecture: n*a(n) +6*(-n+1)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Dec 03 2014
From G. C. Greubel, Jan 09 2023: (Start)
a(n) = [n=0] - Sum_{k=1..n} 2^(n-k)*A000108(k-1).
a(n) = Sum_{j=0..n} 2^(n-j)*A246432(j). (End)

A201634 Triangle read by rows, n>=0, k>=0, T(n,n) = 2^n, T(n,k) = sum_{j=0..k} T(n-1,j) for k=0..n-1.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 8, 8, 1, 5, 13, 21, 16, 1, 6, 19, 40, 56, 32, 1, 7, 26, 66, 122, 154, 64, 1, 8, 34, 100, 222, 376, 440, 128, 1, 9, 43, 143, 365, 741, 1181, 1309, 256, 1, 10, 53, 196, 561, 1302, 2483, 3792, 4048, 512, 1, 11, 64, 260, 821, 2123, 4606

Offset: 0

Peter Luschny, Nov 14 2012

Row sums are A014318.

			[0] [1]
[1] [1, 2]
[2] [1, 3, 4]
[3] [1, 4, 8, 8]
[4] [1, 5, 13, 21, 16]
[5] [1, 6, 19, 40, 56, 32]
[6] [1, 7, 26, 66, 122, 154, 64]
[7] [1, 8, 34, 100, 222, 376, 440, 128]

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A247023 (mirrored).

a201634 n k = a201634_tabl !! n !! k
a201634_row n = a201634_tabl !! n
a201634_tabl = iterate (\xs -> scanl1 (+) xs ++ [2 * last xs]) [1]
-- Reinhard Zumkeller, Sep 20 2014

@CachedFunction
def A201634(n, k):
    if k==n: return 2^n
    return add(A201634(n-1, j) for j in (0..k))
for n in (0..7) : print([A201634(n, k) for k in (0..n)])

A247023 Riordan array (1/(1-2x), xC(x)) where C(x) is the o.g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 21, 13, 5, 1, 32, 56, 40, 19, 6, 1, 64, 154, 122, 66, 26, 7, 1, 128, 440, 376, 222, 100, 34, 8, 1, 256, 1309, 1181, 741, 365, 143, 43, 9, 1, 512, 4048, 3792, 2483, 1302, 561, 196, 53, 10, 1, 1024, 12958, 12446, 8398, 4606

Offset: 0

Philippe Deléham, Sep 13 2014

Row sums are A014318.

Mirror image of the triangle in A201634.

			Triangle begins
1
2, 1
4, 3, 1
8, 8, 4, 1
16, 21, 13, 5, 1
32, 56, 40, 19, 6, 1
64, 154, 122, 66, 26, 7, 1
Production matrix begins
2, 1
0, 1, 1
0, 1, 1, 1
0, 1, 1, 1, 1
0, 1, 1, 1, 1, 1
0, 1, 1, 1, 1, 1, 1
0, 1, 1, 1, 1, 1, 1, 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A000079, A000108, A014318, A201634.

a247023 n k = a247023_tabl !! n !! k
a247023_row n = a247023_tabl !! n
a247023_tabl = map reverse a201634_tabl
-- Reinhard Zumkeller, Sep 20 2014

T(n,0) = 2^n = A000079(n).

T(n+1,1) = A014318(n).

Data corrected by Reinhard Zumkeller, Sep 20 2014

A127156 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n starting with exactly k consecutive pyramids. A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D=(1,-1). This definition differs from the one in A091866.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 5, 2, 3, 3, 1, 19, 7, 5, 6, 4, 1, 67, 26, 12, 11, 10, 5, 1, 232, 93, 38, 23, 21, 15, 6, 1, 804, 325, 131, 61, 44, 36, 21, 7, 1, 2806, 1129, 456, 192, 105, 80, 57, 28, 8, 1, 9878, 3935, 1585, 648, 297, 185, 137, 85, 36, 9, 1, 35072, 13813, 5520

Offset: 0

Emeric Deutsch, Feb 27 2007

Row sums yield the Catalan numbers (A000108). T(n,0)=A114277(n-3) for n>=3. Sum(k*T(n,k), k=0..n)=A014318(n-1) for n>=1.

			T(5,2)=5 because we have (UDUD)UUDUDD, (UUDDUUUDDD), (UUUDDDUUDD), (UDUUUUDDDD) and (UUUUDDDDUD) (the initial 2 pyramids are shown between parentheses).
Triangle starts:
1;
0,1;
0,1,1;
1,1,2,1;
5,2,3,3,1;
19,7,5,6,4,1;

Cf. A000108, A114277, A014318.

C:=(1-sqrt(1-4*z))/2/z: G:=(1-2*z)*C/(1-z-t*z): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(expand(coeff(Gser,z^n))) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.=G(t,z)=(1-2z)C/(1-z-tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,k)=T(n-1,k)+T(n-1,k-1) for n,k>=1.

A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2k)!(x - 1)^(n-k)/((k + 1)!*k!).

Original entry on oeis.org

1, 0, 1, 2, -1, 1, 3, 3, -2, 1, 11, 0, 5, -3, 1, 31, 11, -5, 8, -4, 1, 101, 20, 16, -13, 12, -5, 1, 328, 81, 4, 29, -25, 17, -6, 1, 1102, 247, 77, -25, 54, -42, 23, -7, 1, 3760, 855, 170, 102, -79, 96, -65, 30, -8, 1, 13036, 2905, 685, 68, 181, -175, 161, -95, 38, -9, 1, 45750, 10131, 2220, 617, -113, 356, -336, 256, -133, 47, -10, 1

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

The polynomials C_n(x) have generating function G(x,t) = (1 - sqrt(1 - 4*t))/(2*t*(1 + t - x*t)) = 1 + x*t + (x^2 - x + 2)*t^2 + (x^3 - 2*x^2 + 3*x + 3)*t^3 + ...
C_n(x) can be defined by the recurrence relation C_n(x) = (x - 1)*C_(n-1)(x) + (2n)!/((n + 1)!*n!), C_0(x) = 1 or the equivalent form C_n(x) = (x - 1)*C_(n-1)(x) + C_n(1), C_0(x) = 1.
C_n(x) can be defined as convolution of Catalan numbers and powers of (x - 1).
Discriminants of C_n(x) gives the sequence: 1, 1, -7, -543, 533489, 7080307052, -1318026434480736, -3526797951451513832247, 137992774365121594001729513153, ...
C_n(0) = A032357(n).
C_n(1) = C_n(x) - (x - 1)*C_(n-1)(x) = A000108(n).
C_n(2) = Sum_{m=0..n} C_1(m) = A014137(n).
C_n(3) = A014318(n).
C_n(5) = A000346(n).
C_n(6) = A046714(n).

			Triangle begins:
   1;
   0,  1;
   2, -1,  1;
   3,  3, -2,  1;
  11,  0,  5, -3,  1;
  31, 11, -5,  8, -4,  1;
  ...
The first few polynomials are:
  C_0(x) = 1;
  C_1(x) = x;
  C_2(x) = x^2 -   x   + 2;
  C_3(x) = x^3 - 2*x^2 + 3*x   + 3;
  C_4(x) = x^4 - 3*x^3 + 5*x^2         + 11;
  C_5(x) = x^5 - 4*x^4 + 8*x^3 - 5*x^2 + 11*x + 31;
  ...

G. C. Greubel, Rows n=0..100 of triangle, flattened
Ilya Gutkovskiy, Polynomials C_n(x)
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A130595.

CoefficientList[RecurrenceTable[{c[0] == 1, c[n] == (x - 1) c[n - 1] + CatalanNumber[n]}, c, {n, 11}], x]
T[n_, n_]:= 1; T[n_, 0]:= (-1)^n*Sum[CatalanNumber[k]*(-1)^k, {k, 0, n}]; T[n_, k_]:= T[n - 1, k - 1] - T[n - 1, k]; Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 04 2018 *)

{T(n, k) = if(k==n, 1, if(k==0, sum(j=0,n, (-1)^(n-j)*(2*j)!/(j!*(j+1)!)), T(n-1, k-1) - T(n-1, k))) };
for(n=0, 10, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 04 2018

For triangle: T(n,n)=1, T(n,0) = Sum_{k=0..n} (-1)^(n-k)*(2*k)!/(k! * (k+1)!), T(n, k) = T(n-1, k-1) - T(n-1, k). - G. C. Greubel, Nov 04 2018

cp's OEIS Frontend

A000346 a(n) = 2^(2*n+1) - binomial(2*n+1, n+1).

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A064613 Second binomial transform of the Catalan numbers.

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A097331 Expansion of 1 + 2x/(1 + sqrt(1 - 4x^2)).

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A106270 Inverse of number triangle A106268; triangle T(n,k), 0 <= k <= n.

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A097332 Expansion of (1/(1-x))(1+2x/(1-x+sqrt(1-2x-3x^2))).

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A158495 Expansion of ((1-4*x) + sqrt(1-4*x))/(2*(1-2*x)).

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A201634 Triangle read by rows, n>=0, k>=0, T(n,n) = 2^n, T(n,k) = sum_{j=0..k} T(n-1,j) for k=0..n-1.

A000346 a(n) = 2^(2n+1) - binomial(2n+1, n+1).

A158495 Expansion of ((1-4x) + sqrt(1-4x))/(2(1-2x)).

A247023 Riordan array (1/(1-2x), xC(x)) where C(x) is the o.g.f. of Catalan numbers A000108.

A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2k)!(x - 1)^(n-k)/((k + 1)!*k!).