A005717 -id:A005717 - cp's OEIS frontend

A113682 Expansion of 2/(sqrt(1-2x-3x^2)(1+x+sqrt(1-2x-3*x^2))).

1, 1, 4, 9, 26, 70, 197, 553, 1570, 4476, 12827, 36894, 106471, 308113, 893804, 2598313, 7567466, 22076404, 64498427, 188689684, 552675365, 1620567763, 4756614062, 13974168190, 41088418151, 120906613075, 356035078102

Offset: 0

Paul Barry, Nov 04 2005

Convolution of A002426 and A005043. Diagonal sums of A094531.
Hankel transform is A164611. - Paul Barry, Aug 17 2009
David Scambler observed that [1,0,a(n-2)] for n>=2 count the Dyck paths of semilength n such that the number of peaks equals the number of hills plus the number of returns. - Peter Luschny, Oct 22 2012
Conjectural congruences (working with an offset of 1): a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for prime p >= 5 and positive integers n and k. - Peter Bala, Mar 15 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Cf. A002426, A005043, A094531, A217539, A217540, A246437, A005717.

[(Evaluate(GegenbauerPolynomial(n+1, -n-1), -1/2) + (-1)^n)/2: n in [0..40]];  // G. C. Greubel, Apr 04 2024

ex[x_]:=Module[{sx=Sqrt[1-2x-3x^2]},2/(sx (1+x+sx))]; CoefficientList[ Series[ ex[x],{x,0,40}],x] (* Harvey P. Dale, May 28 2012 *)
Flatten[{1, Table[Coefficient[Sum[(1 + x + x^2)^k, {k, 0, n}], x^n], {n, 1, 30}]}] (* Vaclav Kotesovec, Jan 08 2016 *)

makelist((ultraspherical(n+1,-n-1,-1/2)+(-1)^n)/2,n,0,12); /* Emanuele Munarini, Dec 20 2016 */

x='x+O('x^50); Vec(2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2)))) \\ G. C. Greubel, Feb 28 2017

[(gegenbauer(n+1,-n-1,-1/2) +(-1)^n)/2 for n in range(41)] # G. C. Greubel, Apr 04 2024

a(n) = Sum_{k=0..floor(n/2)} ( Sum_{i=0..n-k} C(n-2k-i, i)*C(n-k, k+i) ).
a(n) = Sum_{k=0..n} A002426(k)*A005043(n-k).
a(n) = Sum_{k=0..n} C(n+1,k+1)*C(k,n-k). - Paul Barry, Aug 21 2007
a(n) = (A002426(n+1) + (-1)^n)/2. - Paul Barry, Aug 17 2009
G.f.: d/dx log(1/(1-x*A005043(x))). - Vladimir Kruchinin, Apr 18 2011
D-finite with recurrence: (n+1)*a(n) +(-n-1)*a(n-1) +(-5*n+1)*a(n-2) +3*(-n+1)*a(n-3)=0. - R. J. Mathar, Nov 26 2012
Recurrence: (n+4)*a(n+3)-(n+4)*a(n+2)-(5*n+14)*a(n+1)-3*(n+2)*a(n)=0. Remark: this recurrence can be obtained using the identity a(n) = (t(n+1)+(-1)^n)/2 and the recurrence of the central trinomial coefficients t(n) = A002426(n). So, the above P-finite recurrences are true. - Emanuele Munarini, Dec 20 2016
a(n) = (-1)^(n+1) * (hypergeom([1/2, -n-1], [1], 4) - 1)/2. - Vladimir Reshetnikov, Apr 25 2016
a(n) = (-1)^n + A246437(n+1). - Vladimir Reshetnikov, Apr 25 2016

A076540 Number of branches in all ordered trees with n edges.

Original entry on oeis.org

1, 3, 11, 41, 154, 582, 2211, 8437, 32318, 124202, 478686, 1849498, 7161556, 27784460, 107980515, 420300045, 1638238710, 6393535170, 24980504010, 97704407790, 382509199020, 1498824792660, 5877754713870, 23067328421826, 90590960500524, 356002519839652

Offset: 1

Emeric Deutsch, Oct 18 2002

Row sums of triangle A136535. - Gary W. Adamson, Jan 04 2008
The average of the n terms a(1),...,a(n) is C(n) = A000108(n), the n-th Catalan number. - Franklin T. Adams-Watters, May 20 2010
Binomial transform of A005717. - Peter Luschny, Jan 17 2012
a(n) is the number of parking functions of size n avoiding the patterns 213, 312, and 321. - Lara Pudwell, Apr 10 2023

			a(3)=11 because the five ordered trees with 3 edges have 1+3+2+2+3 = 11 branches altogether.

Michael De Vlieger, Table of n, a(n) for n = 1..1664
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combinat. Theory, Ser A, 19, 214-222, 1975.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

First differences of A001791. First differences are in A073663.

Cf. A136535, A005717.

[Binomial(2*n,n)+Binomial(2*n,n-1)+Binomial(2*n,n-2): n in [0..30]]; // Vincenzo Librandi, Jun 17 2015

Table[Binomial[2 n, n] + Binomial[2 n, n-1] + Binomial[2 n, n-2], {n, 0, 30}] (* Vincenzo Librandi, Jun 17 2015 *)

vector(30, n, binomial(2*n-1,n-2)+binomial(2*n-2,n-1)) \\ Michel Marcus, Jun 17 2015

a(n) = (3*n^2-2*n+1)*binomial(2*n, n)/(2*(n+1)*(2*n-1)).
G.f.: (1-z)*(C-1)/sqrt(1-4*z), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) = binomial(2n-1, n-2) + binomial(2n-2, n-1). - David Callan, Nov 06 2003
a(n+1) = [x^n](1 + x + x^2)*(1 + x)^(2*n) = binomial(2*n,n) + binomial(2*n,n-1) + binomial(2*n,n-2). - Peter Bala, Jun 15 2015
D-finite with recurrence (n+1)*a(n) +(-7*n+1)*a(n-1) +2*(7*n-12)*a(n-2) +4*(-2*n+7)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

A025178 First differences of the central trinomial coefficients A002426.

Original entry on oeis.org

0, 2, 4, 12, 32, 90, 252, 714, 2032, 5814, 16700, 48136, 139152, 403286, 1171380, 3409020, 9938304, 29017878, 84844044, 248382516, 727971360, 2135784798, 6272092596, 18435108258, 54228499920, 159636389850, 470256930052, 1386170197704

Offset: 1

Clark Kimberling

Previous name was: "a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0 = s(n), |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n), where T is the array defined in A025177."

Note that n-1 divides a(n) for n>=2. - T. D. Noe, Mar 16 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001006, A002426, A005717, A007971, A025177, A079309, A097893, A105696, A162551.

a := n -> 2*(n-1)*hypergeom([1-n/2, 3/2-n/2], [2], 4):
seq(simplify(a(n)), n=1..28); # Peter Luschny, Oct 29 2015

Rest[Differences[CoefficientList[Series[x/Sqrt[1-2x-3x^2],{x,0,30}],x]]] (* Harvey P. Dale, Aug 22 2011 *)
Differences[Table[Hypergeometric2F1[(1-n)/2,1-n/2,1,4],{n,1,29}]] (* Peter Luschny, Nov 03 2015 *)

a(n) = sum(k=1, n\2, binomial(n-1,2*k-1)*binomial(2*k,k)); \\ Altug Alkan, Oct 29 2015

def a():
    b, c, n = 0, 2, 2
    yield b
    while True:
        yield c
        b, c = c, ((2*n-1)*c+3*(n-1)*b)*n//((n+1)*(n-1))
        n += 1
A025178 = a()
print([next(A025178) for  in (1..20)]) # _Peter Luschny, Nov 04 2015

a(n) = T(n,n) for n>=1, where T is the array defined in A025177.
a(n) = A002426(n+1) - A002426(n). - Benoit Cloitre, Nov 02 2002
a(n) is asymptotic to c*3^n/sqrt(n) with c around 1.02... - Benoit Cloitre, Nov 02 2002
a(n) = 2*(n-1)*A001006(n-2). - M. F. Hasler, Oct 24 2011
a(n) = 2*A005717(n-1). - R. J. Mathar, Jul 09 2012
E.g.f. Integral(Integral(2*exp(x)*((1-1/x)*BesselI(1,2*x) + 2*BesselI(0,2*x)))). - Sergei N. Gladkovskii, Aug 16 2012
G.f.: -1/x + (1/x-1)/sqrt(1-2*x-3*x^2). - Sergei N. Gladkovskii, Aug 16 2012
D-finite with recurrence: a(n) = ((2+n)*a(n-2)+3*(3-n)*a(n-3)+3*(n-1)*a(n-1))/n, a(0)=1, a(1)=0, a(2)=2. - Sergei N. Gladkovskii, Aug 16 2012 [adapted to new offset by Peter Luschny, Nov 04 2015]
G.f.: (1-x)/x^2*G(0) - 1/x^2 , where G(k)= 1 + x*(2+3*x)*(4*k+1)/( 4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
From Peter Bala, Oct 28 2015: (Start)
a(n) = Sum_{k = 0..floor(n/2)} binomial(n-1,2*k-1)*binomial(2*k,k). Cf. A097893.
n*(n-2)*a(n) = (2*n-3)*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2) with a(1) = 0, a(2) = 2. (End)
From Peter Luschny, Oct 29 2015: (Start)
a(n) = 2*(n-1)*hypergeom([1-n/2,3/2-n/2],[2],4).
a(n) = (n-1)!*[x^(n-1)](2*exp(x)*BesselI(1,2*x)).
a(n) = (n-1)*A007971(n) for n>=2.
A105696(n) = a(n-1) + a(n) for n>=2.
A162551(n-2) = (1/2)*Sum_{k=1..n} binomial(n,k)*a(k) for n>=2.
A079309(n) = (1/2)*Sum_{k=1..2*n} (-1)^k*binomial(2*n,k)*a(k) for n>=1.
(End)

New name based on a comment by T. D. Noe, Mar 16 2005, offset set to 1 and a(1) = 0 prepended by Peter Luschny, Nov 04 2015

A178834 a(n) counts anti-chains of size two in "0,1,2" Motzkin trees on n edges.

Original entry on oeis.org

0, 0, 1, 5, 23, 91, 341, 1221, 4249, 14465, 48442, 160134, 523872, 1699252, 5472713, 17520217, 55801733, 176942269, 558906164, 1759436704, 5522119250, 17285351782, 53977433618, 168194390290, 523076690018, 1623869984706

Offset: 0

Lifoma Salaam, Dec 27 2010

nonn

"0,1,2" trees are rooted trees where each vertex has outdegree zero, one or two. They are counted by the Motzkin numbers.
From Petros Hadjicostas, Jun 02 2020: (Start)
Let A(r,n) be the number of ordered pairs (T, s), where T is a "0,1,2" tree (Motzkin tree) with n edges and s is an r-element anti-chain in T. See Definition 42, p. 30, in Salaam (2008) but we use different notation here.
An r-element anti-chain in a tree is a set of r nodes such that, for every two nodes u and v in the set, u is neither an ancestor nor a descendant of v.
For the current sequence, a(n) = A(r=2, n) for n >= 0.
Let A[r](z) = Sum_{n >= 0} A(r,n)*z^n be the g.f. of the sequence (A(r,n): n >= 0) for fixed r >= 1.
In Theorem 44 (p. 33), Salaam proved that A[r](z) = c_{r-1} * z^(2*r-2) * L(z)^(r-1) * V(z)^r, where c_r = (1/(r + 1))*binomial(2*r, r) is the r-th Catalan number in A000108, L(z) = T(z) = 1/sqrt(1 - 2*z - 3*z^2) is the g.f. of the central trinomial numbers A002426, and V(z) = T(z)*M(z), where M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2))/(2*z^2) is the g.f. of the Motzkin numbers A001006.
It follows (see Table 2.4, p. 39) that A[r](z) = c_{r-1} * z^(2*r-2) * T(z)^(2*r-1) * M(z)^r for fixed r >= 1.
For r = 1, A[r=1](z) = Sum_{n >= 0} A(r=1, n)*z^n = T(z)*M(z) = V(z) is the g.f. of the total number of vertices in all "0,1,2" trees with n edges (i.e., the g.f. of the sequence (A005717(n+1): n >= 0)).
For r = 2, A[r=2](z) = z^2 * T(z)^3 * M(z)^2 is the g.f. of the current sequence. (End)

			For n = 3, we have a(3) = 5 because there are 5 two-element anti-chains on "0,1,2" Motzkin trees on 3 edges.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Lifoma Salaam, Combinatorial statistics on phylogenetic trees, Ph.D. Dissertation, Howard University, Washington D.C., 2008; see Definition 42 (p. 30), Theorem 44 (p. 33), and Table 2.4 (p. 39).

Cf. A000108, A002426, A001006, A005717.

m:=30; R:=PowerSeriesRing(Rationals(), m); [0,0] cat Coefficients(R!( (1-x-Sqrt(1-2*x-3*x^2))^2/(4*x^2*Sqrt(1-2*x-3*x^2)^3) )); // G. C. Greubel, Jan 21 2019

M:= (1-z -Sqrt[1-2*z-3*z^2])/(2*z^2); T:= 1/Sqrt[1-2*z-3*z^2]; CoefficientList[Series[z^2*M^2*T^3, {z, 0, 30}], z] (* G. C. Greubel, Jan 21 2019 *)

z='z+O('z^33); M=(1-z-sqrt(1-2*z-3*z^2))/(2*z^2); T=1/sqrt(1-2*z-3*z^2); v=Vec(z^2*M^2*T^3+'tmp); v[1]=0; v

((1-x-sqrt(1-2*x-3*x^2))^2/(4*x^2*sqrt(1-2*x-3*x^2)^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 21 2019

G.f.: z^2 * M(z)^2 * T(z)^3, where M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2))/(2*z^2) is the g.f. of the Motzkin numbers and T(z) = 1/sqrt(1 - 2*z - 3*z^2) is the g.f. of the central trinomial numbers.
D-finite with recurrence: -(n-2)*(n+2)*a(n) + (4*n^2-n-8)*a(n-1) + (2*n^2-n-12)*a(n-2) - 3*n*(4*n-3)*a(n-3) - 9*n*(n-1)*a(n-4) = 0. - R. J. Mathar, Jun 14 2016
a(n) ~ 3^(n + 3/2) * sqrt(n) / (4*sqrt(Pi)) * (1 - sqrt(3*Pi)/sqrt(n)). - Vaclav Kotesovec, Mar 08 2023

A372014 T(n,k) is the total number of levels in all Motzkin paths of length n containing exactly k path nodes; triangle T(n,k), n>=0, 1<=k<=n+1, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 4, 6, 4, 3, 1, 8, 14, 12, 7, 4, 1, 18, 32, 33, 21, 11, 5, 1, 44, 74, 84, 64, 34, 16, 6, 1, 113, 180, 208, 181, 111, 52, 22, 7, 1, 296, 457, 520, 485, 344, 179, 76, 29, 8, 1, 782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1

Offset: 0

Alois P. Heinz, Apr 15 2024

A Motzkin path of length n has n+1 nodes.

			In the A001006(3) = 4 Motzkin paths of length 3 there are 2 levels with 1 node, 2 levels with 2 nodes, 2 levels with 3 nodes, and 1 level with 4 nodes.
  2  _     1        1
  2 / \    3 /\_    3 _/\    4 ___    .
  So row 3 is [2, 2, 2, 1].
Triangle T(n,k) begins:
    1;
    0,    1;
    1,    1,    1;
    2,    2,    2,    1;
    4,    6,    4,    3,    1;
    8,   14,   12,    7,    4,   1;
   18,   32,   33,   21,   11,   5,   1;
   44,   74,   84,   64,   34,  16,   6,   1;
  113,  180,  208,  181,  111,  52,  22,   7,  1;
  296,  457,  520,  485,  344, 179,  76,  29,  8, 1;
  782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..28, flattened
Wikipedia, Motzkin number

Columns k=1-2 give: A088457, A051485.
Row sums give A372033 = A001006 + A333498.
Cf. A005717, A371928.

g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)
          , i=0..degree(h)), b(x, y, h)))(p+z^y) end:
b:= proc(x, y, p) option remember; `if`(y+2<=x, g(x-1, y+1, p), 0)
      +`if`(y+1<=x, g(x-1, y, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(n, 0$2)):
seq(T(n), n=0..10);

Sum_{k=1..n+1} k * T(n,k) = A005717(n+1) = (n+1) * A001006(n).

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).

A128017 Expansion of (1+2x+x^2-x^3)/(1-x^2+x^4).

Original entry on oeis.org

1, 2, 2, 1, 1, -1, -1, -2, -2, -1, -1, 1, 1, 2, 2, 1, 1, -1, -1, -2, -2, -1, -1, 1, 1, 2, 2, 1, 1, -1, -1, -2, -2, -1, -1, 1, 1, 2, 2, 1, 1, -1, -1, -2, -2, -1, -1, 1, 1, 2, 2, 1, 1, -1, -1, -2, -2, -1, -1, 1, 1, 2, 2, 1, 1, -1, -1, -2, -2, -1, -1

Offset: 0

Paul Barry, Feb 11 2007

Hankel transform of A128014, A128015, A005717(n+1).

Index entries for linear recurrences with constant coefficients, signature (0,1,0,-1).

Cf. A128016.

A005213 Number of symmetric, reduced unit interval schemes with n+1 intervals (n>=1).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 7, 6, 19, 16, 51, 45, 141, 126, 393, 357, 1107, 1016, 3139, 2907, 8953, 8350, 25653, 24068, 73789, 69576, 212941, 201643, 616227, 585690, 1787607, 1704510, 5196627, 4969152, 15134931, 14508939, 44152809, 42422022, 128996853

Offset: 0

N. J. A. Sloane

nonn

Also, number of symmetric Dyck paths of semilength n with no peaks at odd level. E.g., a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUDUDD, where U=(1,1) and D=(1,-1).

Sequence is obtained by alternating A002426 and A005717.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Phil Hanlon, Counting interval graphs, Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.

Cf. A002426, A005717.

G:=((1+2*z-z^2)/sqrt(1-2*z^2-3*z^4)-1)/(2*z): Gser:=series(G,z=0,40): 1,seq(coeff(Gser,z^n),n=1..38);

CoefficientList[Series[((1 + 2*z - z^2)/Sqrt[1 - 2*z^2 - 3*z^4] - 1)/(2*z), {z, 0, 50}], z] (* G. C. Greubel, Mar 02 2017 *)

x='x +O('x^50); Vec(((1+2*x-x^2)/sqrt(1-2*x^2-3*x^4)-1)/(2*x)) \\ G. C. Greubel, Mar 02 2017

G.f.: ((1+2*z-z^2)/sqrt(1-2*z^2-3*z^4)-1)/(2*z).

a(2*n) = A002426(n), a(2*n+1) = [A002426(n+1) - A002426(n)]/2, (A002426(n) are the central trinomial coefficients).

Edited by Emeric Deutsch, Nov 21 2003

A147657 a(1)=1, a(2)=2, thereafter (1, -2, 3, -4, 5, -6, ...) interleaved with (-2, 2, -2, 2, ...).

Original entry on oeis.org

1, 2, 1, -2, -2, 2, 3, -2, -4, 2, 5, -2, -6, 2, 7, -2, -8, 2, 9, -2, -10, 2, 11, -2, -12, 2, 13, -2, -14, 2, 15, -2, -16, 2, 17, -2, -18, 2, 19, -2, -20, 2, 21, -2, -22, 2, 23, -2, -24, 2, 25, -2, -26, 2, 27, -2, -28, 2, 29, -2, -30, 2, 31, -2, -32, 2, 33

Offset: 1

Gary W. Adamson, Nov 09 2008

Equals POLYMOTZKINTINV [1,2,3,...], such that POLYMOTZKINT A147657 = [1,2,3,...]. A comment accompanying the POLYMOTZKINT operation may be found in A005717.

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A005717, A147658.

Join[{1,2}, LinearRecurrence[{0, -2, 0, -1}, {1, -2, -2, 2}, 65]] (* Georg Fischer, Nov 02 2021 *)

(1, -2, 3, -4, 5,...) interleaved with (1, 2, -2, 2,...) such that the first subsequence starts after the first "2" in the second subsequence.

G.f.: x + x^2*(2+x+2*x^2)/(1+x^2)^2. - R. J. Mathar, Dec 13 2022

Definition and a(28) ff. corrected by Georg Fischer, Nov 02 2021

A147658 (1, 2, -4, 6, -8, ...) interleaved with (3, -3, 3, -3, 3, ...).

Original entry on oeis.org

1, 3, 2, -3, -4, 3, 6, -3, -8, 3, 10, -3, -12, 3, 14, -3, -16, 3, 18, -3, -20, 3, 22, -3, -24, 3, 26, -3, -28, 3, 30, -3, -32, 3, 34, -3, -36, 3, 38, -3, -40, 3, 42, -3, -44, 3, 46, -3, -48, 3, 50, -3, -52, 3, 54, -3, -56, 3, 58, -3, -60, 3, 62, -3, -64

Offset: 1

Gary W. Adamson, Nov 09 2008

POLYMOTZKINT A147657 = [1,2,3,...].
POLYMOTZKINTINV operation on [1,3,5,7,...], such that POLYMOTZKINT A147658 = [1,3,5,7,...].
Cf. A005717 for an example of the POLYMOTZKINT operation.

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A005717, A147657.

with(ListTools): Flatten([1, seq([(-1)^(k-1)*3, (-1)^(k-1)*2*k], k=1..32)]); # Georg Fischer, Nov 02 2021

a(1) = 1; a(2*k) = (-1)^(k-1)*3; a(2*k+1) = (-1)^(k-1)*2*k for k >= 1. - Georg Fischer, Nov 02 2021

a(25) ff. corrected by Georg Fischer, Nov 02 2021

cp's OEIS Frontend

A113682 Expansion of 2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

SageMath

Formula

A076540 Number of branches in all ordered trees with n edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A025178 First differences of the central trinomial coefficients A002426.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A178834 a(n) counts anti-chains of size two in "0,1,2" Motzkin trees on n edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A372014 T(n,k) is the total number of levels in all Motzkin paths of length n containing exactly k path nodes; triangle T(n,k), n>=0, 1<=k<=n+1, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

A113682 Expansion of 2/(sqrt(1-2x-3x^2)(1+x+sqrt(1-2x-3*x^2))).

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