A007317 -id:A007317 - cp's OEIS frontend

A199475 G.f. satisfies A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+2))/(1 - A(x)^2).

1, 2, 7, 34, 195, 1225, 8146, 56336, 401005, 2918308, 21614216, 162385693, 1234515922, 9479336440, 73410868630, 572719097908, 4496923141241, 35509806367132, 281814387290431, 2246608404455588, 17982234787607464, 144458551104066553, 1164342291135424494

Offset: 0

Paul D. Hanna, Nov 08 2011

nonn

Compare to g.f. B(x) of A007317 (binomial transform of Catalan numbers):

B(x) = Sum_{n>=0} x^n * (1 - B(x)^(n+1))/(1 - B(x)).

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 34*x^3 + 195*x^4 + 1225*x^5 +...
where g.f. A = A(x) satisfies the equivalent expressions:
A = 1 + x*(1-A^4)/(1-A^2) + x^2*(1-A^6)/(1-A^2) + x^3*(1-A^8)/(1-A^2) +...
A = 1 + x*(1 + A^2) + x^2*(1 + A^2 + A^4) + x^3*(1 + A^2 + A^4 + A^6) +...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A007317, A171199, A199548, A349289, A349290, A349291, A349292, A349293.

Rest[CoefficientList[InverseSeries[Series[(2*x^2)/(1 + x^2 - Sqrt[1 - 4*x - 2*x^2 + x^4]), {x, 0, 30}], x], x]] (* Vaclav Kotesovec, Jul 30 2021 *)

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, A^(2*k))+x*O(x^n))); polcoeff(A, n)}

{a(n)=local(A=1+x);for(i=1,n,A=1/((1-x)*(1 - x*A^2+x*O(x^n))));polcoeff(A,n)}

{a(n)=polcoeff(1/x*serreverse(2*x^2/(1+x^2-sqrt(1-4*x-2*x^2+x^4+x^3*O(x^n)))),n)}

G.f. satisfies: A(x) = 1/((1-x)*(1 - x*A(x)^2)).
G.f.: A(x) = (1/x)*Series_Reversion( 2*x^2/(1+x^2 - sqrt(1-4*x-2*x^2+x^4)) ).
G.f. satisfies: A(x) = G(x*A(x)) and G(x) = A(x/G(x)) = g.f. of A171199, where G(x) = exp( Sum_{n>=1} [G(x)^n + G(x)^-n]*x^n/n ).
a(n) = 1 + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Jul 25 2021
a(n) ~ sqrt(387 + 35*sqrt(129)) * (35 + 3*sqrt(129))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Jul 30 2021
a(n) = Sum_{k=0..n} binomial(n+k,n-k) * binomial(3*k,k)/(2*k+1). - Seiichi Manyama, Oct 03 2023
D-finite with recurrence 2*n*(2*n+1)*a(n) +3*(-13*n^2+11*n-2)*a(n-1) +(35*n^2-23*n-42)*a(n-2) +(35*n^2-257*n+426)*a(n-3) +3*(-13*n^2+93*n-166)*a(n-4) +2*(n-4)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Feb 10 2024

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

A124733 Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (2,3,3,...) and super- and subdiagonals (1,1,1,...).

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 15, 21, 8, 1, 51, 86, 46, 11, 1, 188, 355, 235, 80, 14, 1, 731, 1488, 1140, 489, 123, 17, 1, 2950, 6335, 5397, 2730, 875, 175, 20, 1, 12235, 27352, 25256, 14462, 5530, 1420, 236, 23, 1, 51822, 119547, 117582, 74172, 32472, 10026, 2151, 306, 26, 1

Offset: 1

Gary W. Adamson & Roger L. Bagula, Nov 05 2006

With a different offset: Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=2*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+3*T(n-1,k)+T(n-1,k+1) for k>=1. - Philippe Deléham, Mar 27 2007
Equals A007318*A039599 (when written as lower triangular matrix). - Philippe Deléham, Jun 16 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
5^n = (n-th row terms) dot (first n+1 odd integers). Example: 5^4 = 625 = (51, 86, 46, 11, 1) dot (1, 3, 5, 7, 9) = (51 + 258 + 230 + 77 + 9) = 625. [Gary W. Adamson, Jun 13 2011]

			Row 3 is (5,5,1) because M[3]=[2,1,0;1,3,1;0,1,3] and M[3]^2=[5,5,1;5,11,6;1,6,10].
Triangle starts:
1;
2, 1;
5, 5, 1;
15, 21, 8, 1;
51, 86, 46, 11, 1;
188, 355, 235, 80, 14, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Shu-Chiuan Chang and Robert Shrock, Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips, Journal of Statistical Physics 137 (2009) 667.

Cf. A110877, A091965, A002212, A007317, A026375 (row sums).

with(linalg): m:=proc(i,j) if i=1 and j=1 then 2 elif i=j then 3 elif abs(i-j)=1 then 1 else 0 fi end: for n from 3 to 11 do A[n]:=matrix(n,n,m): B[n]:=multiply(seq(A[n],i=1..n-1)) od: 1; 2,1; for n from 3 to 11 do seq(B[n][1,j],j=1..n) od; # yields sequence in triangular form
T := (n,k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k,-n+1,3/2) + GegenbauerC(n-k-1,-n+1,3/2)): seq(seq(T(n,k), k=1..n), n=1..10); # Peter Luschny, May 13 2016

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,  T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 2, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

Sum_{k=0..n} (-1)^(n-k)*T(n,k) = (-1)^n. - Philippe Deléham, Feb 27 2007
Sum_{k=0..n} T(n,k)*(2*k+1) = 5^n. - Philippe Deléham, Mar 27 2007
T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,3/2) + GegenbauerC(n-k-1,-n+1,3/2)). - Peter Luschny, May 13 2016
From Peter Bala, Sep 06 2022: (Start)
The following assume the row and column indexing start at 0.
Riordan array (f(x), x*g(x)), where f(x) = ( 1 - sqrt((1 - 5*x)/(1 - x)) )/(2*x) = 1 + 2*x + 5*x^2 + 15*x^3 + 51*x^4 + ... is the o.g.f. of A007317 and g(x) = ( 1 - 3*x - sqrt(1 - 6*x + 5*x^2) )/(2*x^2) = 1 + 3*x + 10*x^2 + 36*x^3 + 137*x^4 + .... See A002212.
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x)*(1 + 3*x + x^2)^n expanded about the point x = 0.
T(n,k) = a(n,k) - a(n,k+1), where a(n,k) = Sum_{j = 0..n} binomial(n,j)* binomial(j,n-k-j)*3^(2*j+k-n). (End)

Edited by N. J. A. Sloane, Dec 04 2006

A091867 Triangle read by rows: T(n,k) = number of Dyck paths of semilength n having k peaks at odd height.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 3, 4, 6, 0, 1, 6, 15, 10, 10, 0, 1, 15, 36, 45, 20, 15, 0, 1, 36, 105, 126, 105, 35, 21, 0, 1, 91, 288, 420, 336, 210, 56, 28, 0, 1, 232, 819, 1296, 1260, 756, 378, 84, 36, 0, 1, 603, 2320, 4095, 4320, 3150, 1512, 630, 120, 45, 0, 1, 1585, 6633, 12760, 15015, 11880, 6930, 2772, 990, 165, 55, 0, 1

Offset: 0

Emeric Deutsch, Mar 10 2004

Number of ordered trees with n edges having k leaves at odd height. Row sums are the Catalan numbers (A000108). T(n,0)=A005043(n). Sum_{k=0..n} k*T(n,k) = binomial(2n-2,n-1).
T(n,k)=number of Dyck paths of semilength n and having k ascents of length 1 (an ascent is a maximal string of consecutive up steps). Example: T(4,2)=6 because we have UdUduud, UduuddUd, uuddUdUd, uudUdUdd, UduudUdd and uudUddUd (the ascents of length 1 are indicated by U instead of u).
T(n,k) is the number of Łukasiewicz paths of length n having k level steps (i.e., (1,0)). A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: T(4,1)=4 because we have HU(2)DD, U(2)HDD, U(2)DHD and U(2)DDH, where H=(1,0), U(1,1), U(2)=(1,2) and D=(1,-1). - Emeric Deutsch, Jan 06 2005
T(n,k) = number of noncrossing partitions of [n] containing k singleton blocks. Also, T(n,k) = number of noncrossing partitions of [n] containing k adjacencies. An adjacency is an occurrence of 2 consecutive integers in the same block (here 1 and n are considered consecutive). In fact, the statistics # singletons and # adjacencies have a symmetric joint distribution.
Exponential Riordan array [e^x*(Bessel_I(0,2x)-Bessel_I(1,2x)),x]. - Paul Barry, Mar 03 2011
T(n,k) is the number of ordered trees having n edges and exactly k nodes with one child. - Geoffrey Critzer, Feb 25 2013
From Tom Copeland, Nov 04 2014: (Start)
Summing the coeff. of the partitions in A134264 for a Lagrange inversion formula (see also A249548) containing (h_1)^k = (1')^k gives this triangle, so this array's o.g.f. H(x,t) = x + t * x^2 + (1 + t^2) * x^3 ... is the inverse of the o.g.f. of A104597 with a sign change, i.e., H^(-1)(x,t) = (x-x^2) / [1 + (t-1)(x-x^2)] = Cinv(x)/[1 + (t-1)Cinv(x)] = P[Cinv(x),t-1] where Cinv(x)= x * (1-x) is the inverse of C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the Catalan numbers A000108, and P(x,t) = x/(1+t*x) with inverse Pinv(x,t) = -P(-x,t) = x/(1-t*x). Therefore,
O.g.f.: H(x,t) = C[Pinv(x,t-1)] = C[P(x,1-t)] = C[x/(1-(t-1)x)] = {1-sqrt[1-4*x/(1-(t-1)x)]}/2 (for A091867). Reprising,
Inverse O.g.f.: H^(-1)(x,t) = x*(1-x) / [1 + (t-1)x(1-x)] = P[Cinv(x),t-1].
From general arguments in A134264, the row polynomials are an Appell sequence with lowering operator d/dt, having the umbral property (p(.,t)+a)^n=p(n,t+a) with e.g.f. = e^(x*t)/w(x), where 1/w(x)= e.g.f. of first column for the Motzkin numbers in A005043. (Mislabeled argument corrected on Jan 31 2016.)
Cf. A124644 (t-shifted polynomials), A026378 (t=-4), A001700 (t=-3), A005773 (t=-2), A126930 (t=-1) and A210736 (t=-1, a(0)=0, unsigned), A005043 (t=0), A000108 (t=1), A007317 (t=2), A064613 (t=3), A104455 (t=4), A030528 (for inverses).
(End)
The sequence of binomial transforms A126930, A005043, A000108, ... in the above comment appears in A126930 and the link therein to a paper by F. Fite et al. on page 42. - Tom Copeland, Jul 23 2016

			T(4,2)=6 because we have (ud)uu(ud)dd, uu(ud)dd(ud), uu(ud)(ud)dd, (ud)(ud)uudd, (ud)uudd(ud) and uudd(ud)(ud) (here u=(1,1), d=(1,-1) and the peaks at odd height are shown between parentheses).
Triangle begins:
   1;
   0,   1;
   1,   0,   1;
   1,   3,   0,   1;
   3,   4,   6,   0,  1;
   6,  15,  10,  10,  0,  1;
  15,  36,  45,  20, 15,  0, 1;
  36, 105, 126, 105, 35, 21, 0, 1;
  ...

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison and Wesley, 1996, page 254 (first edition)

Alois P. Heinz, Rows n = 0..200, flattened
David Callan, On conjugates for set partitions and integer compositions , arXiv:math/0508052 [math.CO], 2005.
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.

Cf. A000108, A001700, A005043, A005773, A007317, A026378, A030528, A064613, A104455, A104597, A124644, A126930, A210736.

T := proc(n,k) if k>n then 0 elif k=n then 1 else (binomial(n+1,k)/(n+1))*sum(binomial(n+1-k,j)*binomial(n-k-j-1,j-1),j=1..floor((n-k)/2)) fi end: seq(seq(T(n,k),k=0..n),n=0..12);
T := (n,k) -> (-1)^(n+k)*binomial(n,k)*hypergeom([-n+k,1/2],[2],4): seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Jul 27 2016
# alternative Maple program:
b:= proc(x, y, t) option remember; expand(`if`(x=0, 1,
      `if`(y>0, b(x-1, y-1, 0)*z^irem(t*y, 2), 0)+
      `if`(y (p-> seq(coeff(p, z, i), i=0..n))(b(2*n, 0$2)):
seq(T(n), n=0..16);  # Alois P. Heinz, May 12 2017

nn=10;cy = ( 1 + x - x y - ( -4x(1+x-x y) + (-1 -x + x y)^2)^(1/2))/(2(1+x-x y)); Drop[CoefficientList[Series[cy,{x,0,nn}],{x,y}],1]//Grid  (* Geoffrey Critzer, Feb 25 2013 *)
Table[Which[k == n, 1, k > n, 0, True, (Binomial[n + 1, k]/(n + 1)) Sum[Binomial[n + 1 - k, j] Binomial[n - k - j - 1, j - 1], {j, Floor[(n - k)/2]}]], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 25 2016 *)

T(n, k) = [binomial(n+1, k)/(n+1)]*Sum_{j=1..floor((n-k)/2)} binomial(n+1-k, j)*binomial(n-k-j-1, j-1) for kn. G.f.=G=G(t, z) satisfies z(1+z-tz)G^2-(1+z-tz)G+1=0. T(n, k)=r(n-k)*binomial(n, k), where r(n)=A005043(n) are the Riordan numbers.
G.f.: 1/(1-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-... (continued fraction). - Paul Barry, Aug 03 2009
Sum_{k=0..n} T(n,k)*x^k = A126930(n), A005043(n), A000108(n), A007317(n), A064613(n), A104455(n) for x = -1,0,1,2,3,4 respectively. - Philippe Deléham, Dec 03 2009
Sum_{k=0..n} (-1)^(n-k)*T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Dec 03 2009
E.g.f.: e^(x+xy)*(Bessel_I(0,2x)-Bessel_I(1,2x)). - Paul Barry, Mar 10 2010
From Tom Copeland, Nov 06 2014: (Start)
O.g.f.: H(x,t) = {1-sqrt[1-4x/(1-(t-1)x)]}/2 (shifted index, as given in Copeland's comment, see comp. inverse there).
H(x,t)= x / [1-(C.+(t-1))x] = Sum_{n>=1} (C.+ (t-1))^(n-1)*x^n umbrally, e.g., (a.+b.)^2 = a_0*b_2 + 2 a_1*b1_+ a_0*b_2, where (C.)^n = C_n are the Catalan numbers (1,1,2,5,14,..) of A000108.
This shows directly that the lowering operator for the polynomials is D=d/dt, i.e., D p(n,t)= D(C. + (t-1))^n = n * (C. + (t-1))^(n-1) = n*p(n-1,t), so that the polynomials form an Appell sequence, and that p(n,0) gives a Motzkin sum, or Riordan, number A005043.
(End)
T(n,k) = (-1)^(n+k)*binomial(n,k)*hypergeom([k-n,1/2],[2],4). - Peter Luschny, Jul 27 2016

A104455 Expansion of e.g.f. exp(5x)(BesselI(0,2x) - BesselI(1,2x)).

Original entry on oeis.org

1, 4, 17, 77, 371, 1890, 10095, 56040, 320795, 1881524, 11250827, 68330773, 420314629, 2612922694, 16389162537, 103587298965, 659071002195, 4217699773140, 27129590096595, 175303621195647, 1137400502295081, 7406899253418414, 48396105031873197, 317180187174490902, 2084542632685363221

Offset: 0

Paul Barry, Mar 08 2005

Third binomial transform of A000108. In general, the k-th binomial transform of A000108 will have g.f. (1-sqrt((1-(k+4)*x)/(1-k*x)))/(2*x), e.g.f. exp((k+2)*x)*(BesselI(0,2*x) - BesselI(1,2*x)) and a(n) = Sum_{i=0..n} C(n,i)*C(i)*k^(n-i).
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
In general, the k-th binomial transform of A000108 can be generated from M^n, M = the production matrix of the form shown in the formula section, with a diagonal (k+1, k+1, k+1, ...). - Gary W. Adamson, Jul 21 2011
a(n) is the number of Schroeder paths of semilength n in which the H=(2,0) steps come in 3 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(1-4*x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf A091867.)
O.g.f.: G(x) = C[P[P[P(x,-1),-1]]-1] = C[P(x,-3)] = [1-sqrt(1-4*x/(1-3*x)]/2 = x*A104455(x).
Ginv(x) =  Pinv[Cinv(x),-3]= P[Cinv(x),3] = x*(1-x)/[1+3*x*(1-x)] = (x-x^2)/[1+3(x-x^2)] = x*A125145(-x). (Cf. A030528.) (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A007317, A064613.

Cf. A000108, A091867, A125145, A030528.

CoefficientList[Series[(1-Sqrt[(1-7*x)/(1-3*x)])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

x='x+O('x^66); Vec((1-sqrt((1-7*x)/(1-3*x)))/(2*x)) \\ Joerg Arndt, Mar 31 2013

G.f.: (1-sqrt((1-7*x)/(1-3*x)))/(2*x).
a(n) = Sum_{k=0..n} C(n, k)*C(k)*3^(n-k).
a(n) = 3^n+Sum_{k=0..n-1} a(k)*a(n-1-k), a(0)=1. - Philippe Deléham, Dec 12 2009
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
  4, 1, 0, 0, ...
  1, 4, 1, 0, ...
  1, 1, 4, 1, ...
  1, 1, 1, 4, ...
(End)
D-finite with recurrence: (n+1)*a(n) = 2*(5*n-1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 7^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
G.f. A(x) satisfies: A(x) = 1/(1 - 3*x) + x * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020

A118376 Number of all trees of weight n, where nodes have positive integer weights and the sum of the weights of the children of a node is equal to the weight of the node.

Original entry on oeis.org

1, 2, 6, 24, 112, 568, 3032, 16768, 95200, 551616, 3248704, 19389824, 117021824, 712934784, 4378663296, 27081760768, 168530142720, 1054464293888, 6629484729344, 41860283723776, 265346078982144, 1687918305128448, 10771600724946944, 68941213290561536

Offset: 1

Jeremy Johnson (jjohnson(AT)cs.drexel.edu), May 15 2006

nonn

The number of trees with leaf nodes equal to 1 is counted by the sequence A001003 of super-Catalan numbers. The number of binary trees is counted by the sequence A007317 and the number of binary trees with leaf nodes equal to 1 is counted by the sequence A000108 of Catalan numbers.
Also the number of series-reduced enriched plane trees of weight n. A series-reduced enriched plane tree of weight n is either the number n itself or a finite sequence of at least two series-reduced enriched plane trees, one of each part of an integer composition of n. For example, the a(3) = 6 trees are: 3, (21), (12), (111), ((11)1), (1(11)). - Gus Wiseman, Sep 11 2018
Conjectured to be the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 3>4, 3>5} of length 5. That is, conjectured to be the number of length n permutations having no subsequences of length 5 in which the first element is the largest, and the third element is larger than the fourth and fifth elements. - Sergey Kitaev, Dec 13 2020
This conjecture has been proven. It can be restated as the number of size n permutations avoiding 51423, 51432, 52413, 52431, 53412, 53421, 54312, 54321. There are twelve sets of permutations avoiding eight size five permutations that are known to match this sequence. A further four are conjectured to match this sequence. - Christian Bean, Jul 24 2024

			T(3) = 6 because there are six trees
  3    3      3     3     3       3
      2 1    2 1   1 2   1 2    1 1 1
            1 1           1 1
From _Gus Wiseman_, Sep 11 2018: (Start)
The a(4) = 24 series-reduced enriched plane trees:
  4,
  (31), (13), (22), (211), (121), (112), (1111),
  ((21)1), ((12)1), (1(21)), (1(12)), (2(11)), ((11)2),
  ((111)1), (1(111)), ((11)(11)), ((11)11), (1(11)1), (11(11)),
  (((11)1)1), ((1(11))1), (1((11)1)), (1(1(11))).
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, PermPAL database.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
Hadrien Cambazard and Nicolas Catusse, Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the Plane, arXiv preprint arXiv:1512.06649 [cs.DS], 2015.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Pawel Hitczenko, Jeremy R. Johnson, and Hung-Jen Huang, Distribution of a class of divide and conquer recurrences arising from the computation of the Walsh-Hadamard transform, Theoretical Computer Science, Vol. 352, 2006, pp. 8-30.
J. R. Johnson and M. Püschel, In search for the optimal Walsh-Hadamard transform, Proc. ICASSP, Vol. 4, 2000, pp. 3347-3350.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A000108, A000311, A001003, A005804, A007317, A007853, A032171, A032200, A143363, A289501, A317852.

T := proc(n) option remember; local C, s, p, tp, k, i; if n = 1 then return 1; else s := 1; for k from 2 to n do C := combinat[composition](n,k); for p in C do tp := map(T,p); s := s + mul(tp[i],i=1..nops(tp)); end do; end do; end if; return s; end;

Rest[CoefficientList[Series[(Sqrt[1-8*x+8*x^2]-1)/(4*x-4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 03 2014 *)
a[n_] := 1+Sum[Binomial[n-1, k-1]*Hypergeometric2F1[2-k, k+1, 2, -1], {k, 2, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
urp[n_]:=Prepend[Join@@Table[Tuples[urp/@ptn],{ptn,Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[urp[n]],{n,7}] (* Gus Wiseman, Sep 11 2018 *)

a(n):=sum((-1)^j*2^(n-j-1)*binomial(n,j)*binomial(2*n-2*j-2,n-2*j-1),j,0,(n-1)/2)/n; /* Vladimir Kruchinin, Sep 29 2020 */

x='x+O('x^25); Vec((sqrt(1-8*x+8*x^2) - 1)/(4*x-4)) \\ G. C. Greubel, Feb 08 2017

Recurrence: T(1) = 1; For n > 1, T(n) = 1 + Sum_{n=n1+...+nt} T(n1)*...*T(nt).
G.f.: (-1+(1-8*z+8*z^2)^(1/2))/(-4+4*z).
From Vladimir Kruchinin, Sep 03 2010: (Start)
O.g.f.: A(x) = A001003(x/(1-x)).
a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A001003(k), n>0. (End)
D-finite with recurrence: n*a(n) + 3*(-3*n+4)*a(n-1) + 4*(4*n-9)*a(n-2) + 8*(-n+3)*a(n-3) = 0. - R. J. Mathar, Sep 27 2013
a(n) ~ sqrt(sqrt(2)-1) * 2^(n-1/2) * (2+sqrt(2))^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 03 2014
From Peter Bala, Jun 17 2015: (Start)
With offset 0, binomial transform of A001003.
O.g.f. A(x) = series reversion of x*(2*x - 1)/(2*x^2 - 1); 2*(1 - x)*A^2(x) - A(x) + x = 0.
A(x) satisfies the differential equation (x - 9*x^2 + 16*x^3 - 8*x^4)*A'(x) + x*(3 - 4*x)*A(x) + x*(2*x - 1) = 0. Extracting coefficients gives Mathar's recurrence above. (End)
a(n) = Sum_{j=0..(n-1)/2} (-1)^j*2^(n-j-1)*C(n,j)*C(2*n-2*j-2,n-2*j-1)/n. - Vladimir Kruchinin, Sep 29 2020

A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4k,k) / (3k + 1).

Original entry on oeis.org

1, 2, 7, 38, 257, 1935, 15505, 129519, 1115061, 9823160, 88121887, 802227794, 7392428009, 68819554003, 646276497617, 6114880542117, 58237420303109, 557850829527246, 5370956411708779, 51947475492561014, 504492516832543885, 4917564488572565160

Offset: 0

Ilya Gutkovskiy, Jul 26 2021

nonn

Binomial transform of A002293.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002293, A007317, A104859, A188687, A226974, A346647, A346648, A346649, A346650.

A346646 := proc(n)
    hypergeom([-n,1/4,1/2,3/4],[2/3,1,4/3],-256/27) ;
    simplify(%) ;
end proc:
seq(A346646(n),n=0..40) ; # R. J. Mathar, Jan 10 2023

Table[Sum[Binomial[n, k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 21; CoefficientList[Series[Sum[(Binomial[4 k, k]/(3 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {2/3, 1, 4/3}, -256/27], {n, 0, 21}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(4*k,k)/(3*k + 1)); \\ Michel Marcus, Jul 26 2021

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^2 * A(x)^4.
G.f.: Sum_{k>=0} ( binomial(4*k,k) / (3*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 283^(n + 3/2) / (2048 * sqrt(2*Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) -2*(2*n-1) *(91*n^2 -91*n +24)*a(n-1) +6*(n-1) *(155*n^2 -310*n +167)*a(n-2) -438*(n-1) *(n-2)*(2*n-3) *a(n-3) +283*(n-1)*(n-2) *(n-3)*a(n-4)=0. - R. J. Mathar, Aug 17 2023

A346647 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5k,k) / (4k + 1).

Original entry on oeis.org

1, 2, 8, 54, 460, 4361, 43988, 462580, 5014252, 55624944, 628432101, 7205500484, 83632219892, 980710882430, 11601345881748, 138278231052451, 1659037424218780, 20020306637339944, 242835190201382648, 2958961154058610552, 36203518795424475661

Offset: 0

Ilya Gutkovskiy, Jul 26 2021

nonn

Binomial transform of A002294.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002294, A007317, A188687, A227035, A346646, A346648, A346649, A346650.

A346647 := proc(n)
    hypergeom([-n,1/5,2/5,3/5,4/5],[1/2,3/4,1,5/4],-3125/256) ;
    simplify(%) ;
end proc:
seq(A346647(n),n=0..40) ; # R. J. Mathar, Jan 10 2023

Table[Sum[Binomial[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^3 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 20; CoefficientList[Series[Sum[(Binomial[5 k, k]/(4 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5, -n}, {1/2, 3/4, 1, 5/4}, -3125/256], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(5*k,k)/(4*k + 1)); \\ Michel Marcus, Jul 26 2021

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^3 * A(x)^5.
G.f.: Sum_{k>=0} ( binomial(5*k,k) / (4*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 3381^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence +8*n*(4*n+1) *(2*n-1)*(4*n-1)*a(n) +(-4405*n^4 +9322*n^3 -7655*n^2 +2978*n -480)*a(n-1) +12*(n-1) *(1255*n^3 -3829*n^2 +4204*n -1640) *a(n-2) -2*(n-1) *(n-2) *(10655*n^2 -32221*n +26076) *a(n-3) +4*(n-1) *(n-2) *(n-3)*(3445*n -6922) *a(n-4) -3381*(n-1)*(n-2) *(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 17 2023

A026376 a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=2; also a(n) = T(2n,n-1).

Original entry on oeis.org

1, 6, 30, 144, 685, 3258, 15533, 74280, 356283, 1713690, 8263596, 39938616, 193419915, 938430990, 4560542550, 22195961280, 108171753355, 527816696850, 2578310320610, 12607504827600, 61706212037295, 302275142049870, 1481908332595625, 7270432009471224

Offset: 1

Clark Kimberling

Number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0), with exactly one peak at an even level. E.g., a(2)=6 because we have UUDDH, HUUDD, UDUUDD, UUDDUD, UUDHD and UHUDD. - Emeric Deutsch, Dec 28 2003
Number of left steps in all skew Dyck paths of semilength n+1. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=6 because in the 10 (=A002212(3)) skew Dyck paths of semilength 3 ( namely UDUUDL, UUUDLD, UUDUDL, UUUDDL, UUUDLL and five Dyck paths that have no left steps) we have altogether 6 left steps. - Emeric Deutsch, Aug 05 2007
From Gary W. Adamson, May 17 2009: (Start)
Equals A026378 (1, 4, 17, 75, ...) convolved with A007317 (1, 2, 5, 15, 51, ...).
Equals A081671 (1, 3, 11, 45, ...) convolved with A002212 (1, 3, 10, 36, 137, ...).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
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. 18.
Toufik Mansour and José Luis Ramírez, Enumeration of Fuss-skew paths, Ann. Math. Inform. (2022) Vol. 55, 125-136. See p. 129.

Cf. A006318, A002212, A081671, A007317, A026378.

a := n -> simplify(GegenbauerC(n-1, -n, -3/2)):
seq(a(n), n=1..24); # Peter Luschny, May 09 2016

Rest[CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 13 2014 *)

a(n)=if(n<0,0,polcoeff((1+3*x+x^2)^n,n-1))

A026376 = lambda n : n*hypergeometric([1, 3/2, 1-n], [1, 3], -4)
[round(A026376(n).n(100)) for n in (1..24)] # Peter Luschny, Sep 16 2014

# Recurrence:
def A026376():
    x, y, n = 1, 1, 1
    while True:
        x, y = y, ((6*n + 3)*y - (5*n - 5)*x) / (n + 2)
        yield n*x
        n += 1
a = A026376()
[next(a) for i in (1..24)] # Peter Luschny, Sep 16 2014

E.g.f.: exp(3x)*I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 09 2002
G.f.: (1 - 3*z - t)/(2*z*t) where t = sqrt(1-6*z+5*z^2). - Emeric Deutsch, May 25 2003
a(n) = [t^(n+1)](1+3t+t^2)^n. a := n -> Sum_{j=ceiling((n+1)/2)..n} 3^(2j-n-1)*binomial(n, j)*binomial(j, n+1-j). - Emeric Deutsch, Jan 30 2004
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2k, k+1). - Paul Barry, Sep 20 2004
a(n) = n*A002212(n). - Emeric Deutsch, Aug 05 2007
D-finite with recurrence (n+1)*a(n) - 9*n*a(n-1) + (23*n-27)*a(n-2) + 15*(-n+2)*a(n-3) = 0. - R. J. Mathar, Dec 02 2012
a(n) ~ 5^(n+1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 13 2014
a(n) = n*hypergeometric([1, 3/2, 1-n],[1, 3],-4). - Peter Luschny, Sep 16 2014
a(n) = GegenbauerC(n-1, -n, -3/2). - Peter Luschny, May 09 2016

A055879 Least nondecreasing sequence with a(1) = 1 and Hankel transform {1,1,1,1,...}.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 15, 15, 51, 51, 188, 188, 731, 731, 2950, 2950, 12235, 12235, 51822, 51822, 223191, 223191, 974427, 974427, 4302645, 4302645, 19181100, 19181100, 86211885, 86211885, 390248055, 390248055, 1777495635, 1777495635, 8140539950, 8140539950

Offset: 1

John W. Layman, Jul 15 2000

nonn

Hankel transform {t(n)} of {a(n)} is given by t(n) = Det[{a(1), a(2), ..., a(n)}, {a(2), a(3), ..., a(n+1)}, ..., {a(n), a(n+1), ..., a(2n-1)}].
The bisections of this sequence appear to be the binomial transform of the Catalan numbers, A007317. If that is true then the g.f. for this sequence is (1/(2*x))*( 1 + x - (1-x)^(-1)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2)), which occurs in the Cyvin et al. reference.
Self-convolution yields A039658 (shifted left), which is related to enumeration of edge-rooted catafusenes. - Paul D. Hanna, Aug 08 2008

			G.f.: x + x^2 + 2*x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 15*x^7 + 15*x^8 + 51*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..200
B. N. Cyvin et al., A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337 (see V(x)).
S. J. Cyvin et al, Enumeration and classification of Benezenoid systmes. 32. Normal Perifuses with two internal vertices, J. Chem. Inf. Comput. Sci. 32 (1992) 532-540, Table 1.
S. J. Cyvin et al., Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180. See Table 1 second column on page 1174.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A007317, A039658.

a[ n_] := If[ n < 1, 0, With[ {m = n - 1}, SeriesCoefficient[ Nest[ 1 / (1 - x - x^2 / (1 + x - x^2 #)) &, 1, Quotient[ m + 1, 2]], {x, 0, m}]]]; (* Michael Somos, Jul 01 2011 *)
CoefficientList[Series[Sqrt[(1 + x) (1 - 3 x^2 - Sqrt[1 - 6 x^2 + 5 x^4])/(2 (1 - x))]/x^2, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2014 *)

a(n)=n--; local(A=1+x+x*O(x^n));for(i=0,n,B=subst(A,x,-x);A=1+x*A+x^2*A*B);polcoeff(A,n)

a(n)=n++; polcoeff(sqrt((1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4 +x^4*O(x^n)))/(2*(1-x))),n) \\ Paul D. Hanna, Aug 08 2008

{a(n) = local(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n))}; /* Michael Somos, Jul 01 2011 */

G.f.: A(x) = sqrt( (1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*(1-x)) ). G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)*A(-x). - Paul D. Hanna, Aug 08 2008
G.f.: 1/(1-x-x^2/(1+x-x^2/(1-x-x^2/(1+x-x^2/(1-... (continued fraction). - Paul Barry, Feb 11 2009
D-finite with recurrence (n+1)*a(n) - a(n-1) + (-6*n+11)*a(n-2) + 5*a(n-3) + 5*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012
G.f.: sqrt((1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*(1-x)))/x. - Vaclav Kotesovec, Feb 13 2014
a(n) ~ (5+sqrt(5) - (-1)^n*(5-sqrt(5))) * sqrt(2) * 5^(n/2) / (8 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014
a(n) = a(n-1) if n is even. a(n) = a(n-1)+A002212((n-1)/2) if n is odd. [Cyvin (1992) eq (14)] - R. J. Mathar, Dec 15 2020

More terms from Vincenzo Librandi, Feb 14 2014

cp's OEIS Frontend

A199475 G.f. satisfies A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+2))/(1 - A(x)^2).

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A064613 Second binomial transform of the Catalan numbers.

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A124733 Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (2,3,3,...) and super- and subdiagonals (1,1,1,...).

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A091867 Triangle read by rows: T(n,k) = number of Dyck paths of semilength n having k peaks at odd height.

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A104455 Expansion of e.g.f. exp(5*x)*(BesselI(0,2*x) - BesselI(1,2*x)).

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A118376 Number of all trees of weight n, where nodes have positive integer weights and the sum of the weights of the children of a node is equal to the weight of the node.

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A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) / (3*k + 1).

A104455 Expansion of e.g.f. exp(5x)(BesselI(0,2x) - BesselI(1,2x)).

A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4k,k) / (3k + 1).

A346647 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5k,k) / (4k + 1).