A030528 -id:A030528 - cp's OEIS frontend

A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

1, -2, 1, 3, -4, 1, -4, 10, -6, 1, 5, -20, 21, -8, 1, -6, 35, -56, 36, -10, 1, 7, -56, 126, -120, 55, -12, 1, -8, 84, -252, 330, -220, 78, -14, 1, 9, -120, 462, -792, 715, -364, 105, -16, 1, -10, 165, -792, 1716, -2002, 1365, -560, 136, -18, 1, 11, -220, 1287, -3432, 5005, -4368, 2380, -816, 171, -20

Offset: 0

Wolfdieter Lang

Apart from signs, identical to A078812.
Another version with row-leading 0's and differing signs is given by A285072.
G.f. for row polynomials S(n,x-2) (signed triangle): 1/(1+(2-x)*z+z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-(2+x)*z+z^2) for row polynomials.
Row sums (signed triangle) A049347(n) (periodic(1,-1,0)). Row sums (unsigned triangle) A001906(n+1)=F(2*(n+1)) (even-indexed Fibonacci).
In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The (unsigned) column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for m=0..5, resp. For m=6..23 they are A010966..(+2)..A011000 and for m=24..49 they are A017713..(+2)..A017763.
Riordan array (1/(1+x)^2,x/(1+x)^2). Inverse array is A039598. Diagonal sums have g.f. 1/(1+x^2). - Paul Barry, Mar 17 2005. Corrected by Wolfdieter Lang, Nov 13 2012.
Unsigned version is in A078812. - Philippe Deléham, Nov 05 2006
Also row n gives (except for an overall sign) coefficients of characteristic polynomial of the Cartan matrix for the root system A_n. - Roger L. Bagula, May 23 2007
From Wolfdieter Lang, Nov 13 2012: (Start)
The A-sequence for this Riordan triangle is A115141, and the Z-sequence is A115141(n+1), n>=0. For A- and Z-sequences for Riordan matrices see the W. Lang link under A006232 with details and references.
S(n,x^2-2) = sum(r(j,x^2),j=0..n) with Chebyshev's S-polynomials and r(j,x^2) := R(2*j+1,x)/x, where R(n,x) are the monic integer Chebyshv T-polynomials with coefficients given in A127672. Proof from comparing the o.g.f. of the partial sum of the r(j,x^2) polynomials (see a comment on the signed Riordan triangle A111125) with the present Riordan type o.g.f. for the row polynomials with x -> x^2.  (End)
S(n,x^2-2) = S(2*n+1,x)/x, n >= 0, from the odd part of the bisection of the o.g.f. - Wolfdieter Lang, Dec 17 2012
For a relation to a generator for the Narayana numbers A001263, see A119900, whose columns are unsigned shifted rows (or antidiagonals) of this array, referring to the tables in the example sections. - Tom Copeland, Oct 29 2014
The unsigned rows of this array are alternating rows of a mirrored A011973 and alternating shifted rows of A030528 for the Fibonacci polynomials. - Tom Copeland, Nov 04 2014
Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment in A046521 with references): a(n, m) =  (2*(m + 1)/(n - m))*Sum_{k = m..n-1} (-1)^(n-k)*a(k, m), with input a(n, n) = 1, and a(n,k) = 0 for n < k. - Wolfdieter Lang, Jun 03 2020
Row n gives the characteristic polynomial of the (n X n)-matrix M where M[i,j] = 2 if i = j, -1 if |i-j| = 1 and 0 otherwise. The matrix M is positive definite and has 2-condition number (cot(Pi/(2*n+2)))^2. - Jianing Song, Jun 21 2022
Also the convolution triangle of (-1)^(n+1)*n. - Peter Luschny, Oct 07 2022

			The triangle a(n,m) begins:
n\m   0    1    2     3     4     5     6    7    8  9
0:    1
1:   -2    1
2:    3   -4    1
3:   -4   10   -6     1
4:    5  -20   21    -8     1
5:   -6   35  -56    36   -10     1
6:    7  -56  126  -120    55   -12     1
7:   -8   84 -252   330  -220    78   -14    1
8:    9 -120  462  -792   715  -364   105  -16    1
9:  -10  165 -792  1716 -2002  1365  -560  136  -18  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 13 2012
E.g., fourth row (n=3) {-4,10,-6,1} corresponds to the polynomial S(3,x-2) = -4+10*x-6*x^2+x^3.
From _Wolfdieter Lang_, Nov 13 2012: (Start)
Recurrence: a(5,1) = 35 = 1*5 + (-2)*(-20) -1*(10).
Recurrence from Z-sequence [-2,-1,-2,-5,...]: a(5,0) = -6 = (-2)*5 + (-1)*(-20) + (-2)*21 + (-5)*(-8) + (-14)*1.
Recurrence from A-sequence [1,-2,-1,-2,-5,...]: a(5,1) = 35 = 1*5  + (-2)*(-20) + (-1)*21 + (-2)*(-8) + (-5)*1.
(End)
E.g., the fourth row (n=3) {-4,10,-6,1} corresponds also to the polynomial S(7,x)/x = -4 + 10*x^2 - 6*x^4 + x^6. - _Wolfdieter Lang_, Dec 17 2012
Boas-Buck type recurrence: -56 = a(5, 2) = 2*(-1*1 + 1*(-6) - 1*21) = -2*28 = -56. - _Wolfdieter Lang_, Jun 03 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62.
Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S.: ISBN 0-8218-2848-7, 1978, p. 463.

T. D. Noe, Rows n=0..50 of triangle, flattened
Wolfdieter Lang, First rows of the triangle.
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].
P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Naiomi T. Cameron and Asamoah Nkwanta, On some (pseudo) involutions in the Riordan group, J. of Integer Sequences, 8(2005), 1-16.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - _Tom Copeland_, Oct 11 2014
Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
Eric Weisstein's World of Mathematics, Cartan Matrix
Eric Weisstein's World of Mathematics, Dynkin Diagram
Index entries for sequences related to Chebyshev polynomials.

Cf. A005248, A127677, A053123, A049310.
Cf. A011973, A030528, A034867, A119900.
Cf. A285072 (version with row-leading 0's and differing signs). - Eric W. Weisstein, Apr 09 2017

seq(seq((-1)^(n+m)*binomial(n+m+1,2*m+1),m=0..n),n=0..10); # Robert Israel, Oct 15 2014
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> -(-1)^n*n); # Peter Luschny, Oct 07 2022

T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] (* Roger L. Bagula, May 23 2007 *)
(* Alternative code for the matrices from MathWorld: *)
sln[n_] := 2IdentityMatrix[n] - PadLeft[PadRight[IdentityMatrix[n - 1], {n, n - 1}], {n, n}] - PadLeft[PadRight[IdentityMatrix[n - 1], {n - 1, n}], {n, n}] (* Roger L. Bagula, May 23 2007 *)

@CachedFunction
def A053122(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A053122(n-1,k-1)-A053122(n-2,k)-2*A053122(n-1,k)
for n in (0..9): [A053122(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

a(n, m) := 0 if n

a(n, m) = -2*a(n-1, m) + a(n-1, m-1) - a(n-2, m), a(n, -1) := 0 =: a(-1, m), a(0, 0)=1, a(n, m) := 0 if n

O.g.f. for m-th column (signed triangle): ((x/(1+x)^2)^m)/(1+x)^2.

From Jianing Song, Jun 21 2022: (Start)

T(n,k) = [x^k]f_n(x), where f_{-1}(x) = 0, f_0(x) = 1, f_n(x) = (x-2)*f_{n-1}(x) - f_{n-2}(x) for n >= 2.

f_n(x) = (((x-2+sqrt(x^2-4*x))/2)^(n+1) - ((x-2-sqrt(x^2-4*x))/2)^(n+1))/sqrt(x^2-4x).

The roots of f_n(x) are 2 + 2*cos(k*Pi/(n+1)) = 4*(cos(k*Pi/(2*n+2)))^2 for 1 <= k <= n. (End)

A146559 Expansion of (1-x)/(1 - 2x + 2x^2).

Original entry on oeis.org

1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32, -64, -64, 0, 128, 256, 256, 0, -512, -1024, -1024, 0, 2048, 4096, 4096, 0, -8192, -16384, -16384, 0, 32768, 65536, 65536, 0, -131072, -262144, -262144, 0, 524288, 1048576, 1048576, 0, -2097152, -4194304

Offset: 0

Philippe Deléham, Nov 01 2008

Partial sums of this sequence give A099087. - Philippe Deléham, Dec 01 2008
From Philippe Deléham, Feb 13 2013, Feb 20 2013: (Start)
Terms of the sequence lie along the right edge of the triangle
  (1)
     (1)
   2    (0)
      2   (-2)
   4     0   (-4)
      4    -4   (-4)
   8     0    -8    (0)
      8    -8    -8    (8)
  16     0   -16     0   (16)
     16   -16   -16    16   (16)
  32     0   -32     0    32    (0)
     32   -32   -32    32    32  (-32)
  64     0   -64     0    64     0  (-64)
  ...
Row sums of triangle are in A104597.
(1+i)^n = a(n) + A009545(n)*i where i = sqrt(-1). (End)
From Tom Copeland, Nov 08 2014: (Start)
This array is a member of a Catalan family (A091867) related by compositions of C(x)= (1-sqrt(1-4*x))/2, an o.g.f. for the Catalan numbers A000108, its inverse Cinv(x) = x(1-x), and the special linear fractional (Möbius) transformation P(x,t) = x / (1+t*x) with inverse P(x,-t) in x.
O.g.f.: G(x) = P[P[Cinv(x),-1],-1] = P[Cinv(x),-2] = x*(1-x)/(1 - 2*x*(1-x)) = x*A146599(x).
Ginv(x) = C[P(x,2)] = (1 - sqrt(1-4*x/(1+2*x)))/2 = x*A126930(x).
G(-x) = -(x*(1+x) - 2*(x*(1+x))^2 + 2^2*(x*(1+x))^3 - ...), and so this array contains the -row sums of A030528 * Diag(1, (-2)^1, 2^2, (-2)^3, ...).
The inverse of -G(-x) is -C[-P(x,-2)]= (-1 + sqrt(1+4*x/(1-2*x)))/2, an o.g.f. for A210736 with a(0) set to zero there. (End)
{A146559, A009545} is the difference analog of {cos(x), sin(x)}. (Cf. the Shevelev link.) - Vladimir Shevelev, Jun 08 2017

			G.f. = 1 + x - 2*x^3 - 4*x^4 - 4*x^5 + 8*x^7 + 16*x^8 + 16*x^9 - 32*x^11 - 64*x^12 - ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Beata Bajorska-Harapińska, Barbara Smoleń and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 21.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (2,-2).

Cf. A000108, A009116, A009545, A030528, A091867, A098158, A099087.

Cf. A104597, A124182, A126930, A121625, A146559, A146599, A210736.

I:=[1,1,0]; [n le 3 select I[n] else 2*Self(n-1)-2*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

G(x):=exp(x)*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..44 ); # Zerinvary Lajos, Apr 05 2009
seq(2^(n/2)*cos(Pi*n/4), n=0..44); # Peter Luschny, Oct 09 2021

CoefficientList[Series[(1-x)/(1-2x+2x^2),{x,0,50}],x] (* or *) LinearRecurrence[{2,-2},{1,1},50] (* Harvey P. Dale, Oct 13 2011 *)

Vec((1-x)/(1-2*x+2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

def A146559(n): return ((1, 1, 0, -2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1) # Chai Wah Wu, Feb 16 2024

def A146559():
    x, y = -1, 0
    while True:
        yield -x
        x, y = x - y, x + y
a = A146559(); [next(a) for i in range(51)]  # Peter Luschny, Jul 11 2013

def A146559(n): return 2^(n/2)*chebyshev_T(n, 1/sqrt(2))
[A146559(n) for n in range(51)] # G. C. Greubel, Apr 17 2023

a(0) = 1, a(1) = 1, a(n) = 2*a(n-1) - 2*a(n-2) for n>1.
a(n) = Sum_{k=0..n} A124182(n,k)*(-2)^(n-k).
a(n) = Sum_{k=0..n} A098158(n,k)*(-1)^(n-k). - Philippe Deléham, Nov 14 2008
a(n) = (-1)^n*A009116(n). - Philippe Deléham, Dec 01 2008
E.g.f.: exp(x)*cos(x). - Zerinvary Lajos, Apr 05 2009
E.g.f.: cos(x)*exp(x) = 1+x/(G(0)-x) where G(k)=4*k+1+x+(x^2)*(4*k+1)/((2*k+1)*(4*k+3)-(x^2)-x*(2*k+1)*(4*k+3)/( 2*k+2+x-x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
a(n) = Re( (1+i)^n ) where i=sqrt(-1). - Stanislav Sykora, Jun 11 2012
G.f.: 1 / (1 - x / (1 + x / (1 - 2*x))) = 1 + x / (1 + 2*x^2 / (1 - 2*x)). - Michael Somos, Jan 03 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(m+k) = a(m)*a(k) - A009545(m)*A009545(k). - Vladimir Shevelev, Jun 08 2017
a(n) = 2^(n/2)*cos(Pi*n/4). - Peter Luschny, Oct 09 2021
a(n) = 2^(n/2)*ChebyshevT(n, 1/sqrt(2)). - G. C. Greubel, Apr 17 2023
From Chai Wah Wu, Feb 15 2024: (Start)
a(n) = Sum_{n=0..floor(n/2)} binomial(n,2j)*(-1)^j = A121625(n)/n^n.
a(n) = 0 if and only if n == 2 mod 4.
(End)

A026729 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jan 19 2003

The signed triangular matrix T(n,k)*(-1)^(n-k) is the inverse matrix of the triangular Catalan convolution matrix A106566(n,k), n=k>=0, with A106566(n,k) = 0 if nPhilippe Deléham, Aug 01 2005
As a number triangle: unsigned version of A109466. - Philippe Deléham, Oct 26 2008
A063967*A130595 as infinite lower triangular matrices. - Philippe Deléham, Dec 11 2008
Modulo 2, this sequence becomes A106344. - Philippe Deléham, Dec 18 2008
Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n) = Sum_{i=0..k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A111808. For example, s_1(n) = binomial(n,1) = n is the first column of A111808 for n>1, s_2(n) = binomial(n,1) + binomial(n,2) is the second column of A111808 for n>1, etc. Therefore, in cases k=3,4,5,6,7,8, s_k(n) is A005581(n), A005712(n), A000574(n), A005714(n), A005715(n), A005716(n), respectively. Besides, s_k(n+5) = A064054(n). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
As a triangle, T(n,k) = binomial(k,n-k). - Peter Bala, Nov 27 2015
For all n >= 0, k >= 0, the k-th homology group of the n-torus H_k(T^n) is the free abelian group of rank T(n,k) = binomial(n,k). See the Math Stack Exchange link below. - Jianing Song, Mar 13 2023

			Array begins
  1 0 0 0 0 0 ...
  1 1 0 0 0 0 ...
  1 2 1 0 0 0 ...
  1 3 3 1 0 0 ...
  1 4 6 4 1 0 ...
As a triangle, this begins
  1
  0 1
  0 1 1
  0 0 2 1
  0 0 1 3 1
  0 0 0 3 4 1
  0 0 0 1 6 5 1
  ...
Production array is
  0    1
  0    1   1
  0   -1   1   1
  0    2  -1   1  1
  0   -5   2  -1  1  1
  0   14  -5   2 -1  1  1
  0  -42  14  -5  2 -1  1  1
  0  132 -42  14 -5  2 -1  1  1
  0 -429 132 -42 14 -5  2 -1  1  1
  ... (Cf. A000108)

Muniru A Asiru, Rows n=0..50 of triangle, flattened
Tom Copeland, Addendum to Elliptic Lie Triad
Math Stack Exchange, Homology of the n-torus using the Künneth Formula
Lili Mu and Sai-nan Zheng, On the Total Positivity of Delannoy-Like Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

The official entry for Pascal's triangle is A007318. See also A052553 (the same array read by upward antidiagonals).
Cf. A030528 (subtriangle for 1<=k<=n).
Cf. A109466, A102426.

nmax:=15;; T:=List([0..nmax],n->List([0..nmax],k->Binomial(n,k)));;
b:=List([2..nmax],n->OrderedPartitions(n,2));;
a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][1]][b[i][j][2]]))); # Muniru A Asiru, Jul 17 2018

/* As triangle: */ [[Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 29 2015

seq(seq(binomial(k,n-k),k=0..n),n=0..12); # Peter Luschny, May 31 2014

Table[Binomial[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 28 2015 *)

As a number triangle, this is defined by T(n,0) = 0^n, T(0,k) = 0^k, T(n,k) = T(n-1,k-1) + Sum_{j, j>=0} (-1)^j*T(n-1,k+j)*A000108(j) for n>0 and k>0. - Philippe Deléham, Nov 07 2005
As a triangle read by rows, it is [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 22 2006
As a number triangle, this is defined by T(n, k) = Sum_{i=0..n} (-1)^(n+i)*binomial(n, i)*binomial(i+k, i-k) and is the Riordan array ( 1, x*(1+x) ). The row sums of this triangle are F(n+1). - Paul Barry, Jun 21 2004
Sum_{k=0..n} x^k*T(n,k) = A000007(n), A000045(n+1), A002605(n), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for n=0,1,2,3,4,5,6,7,8,9,10. - Philippe Deléham, Oct 16 2006
T(n,k) = A109466(n,k)*(-1)^(n-k). - Philippe Deléham, Dec 11 2008
G.f. for the triangular interpretation: -1/(-1+x*y+x^2*y). - R. J. Mathar, Aug 11 2015
For T(0,0) = 0, the triangle below has the o.g.f. G(x,t) = [t*x(1+x)]/[1-t*x(1+x)]. See A109466 for a signed version and inverse, A030528 for reverse and A102426 for a shifted version. - Tom Copeland, Jan 19 2016

A052952 a(n) = Fibonacci(n+2) - (1-(-1)^n)/2.

Original entry on oeis.org

1, 1, 3, 4, 8, 12, 21, 33, 55, 88, 144, 232, 377, 609, 987, 1596, 2584, 4180, 6765, 10945, 17711, 28656, 46368, 75024, 121393, 196417, 317811, 514228, 832040, 1346268, 2178309, 3524577, 5702887, 9227464, 14930352, 24157816, 39088169, 63245985, 102334155

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equals row sums of triangle A173284. - Gary W. Adamson, Feb 14 2010
The Kn21 sums (see A180662 for definition) of the 'Races with Ties' triangle A035317 produce this sequence. - Johannes W. Meijer, Jul 20 2011
a(n-1), for n >= 1, gives the number of compositions of n with relative prime parts, and parts not exceeding 2. See the row sums of triangle A030528 where for even n the leading 1 is missing. - Wolfdieter Lang, Jul 27 2023

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 33*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1023
K. Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1, eq (1).
Steven Linton, James Propp, Tom Roby, and Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.
H. Ohtsuka and S. Nakamura, On the sum of reciprocal sums of Fibonacci numbers, Fibonacci Quart. 46/47 (2008/2009), 153-159.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A062114, A173284, A059841, A014217, A180662, A035317.
Partial sums of A008346, first differences of A129696.
Cf. also A000032, A000045, A030528.
Cf. A001595, A054450, A066983, A074331, A168309.

List([0..40], n-> Fibonacci(n+2) -(1-(-1)^n)/2); # G. C. Greubel, Jul 10 2019

a052952 n = a052952_list !! n
a052952_list = 1 : 1 : zipWith (+)
   a059841_list (zipWith (+) a052952_list $ tail a052952_list)
-- Reinhard Zumkeller, Jan 06 2012

[Fibonacci(n+2)-(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Dec 02 2016

A052952 :=proc(n)
    option remember;
    local t1;
    if n <= 1 then
        return 1 ;
    fi:
    if n mod 2 = 1 then
        t1:=0
    else
        t1:=1;
    fi:
    procname(n-1)+procname(n-2)+t1;
end proc;
seq(A052952(n), n=0..40) ; # N. J. A. Sloane, May 25 2008

Table[Fibonacci[n+2] -(1-(-1)^n)/2, {n, 0, 40}] (* Vincenzo Librandi, Dec 02 2016 *)
Sum[(-1)^k*Fibonacci[Range[2,41], 1-k], {k,0,1}] (* G. C. Greubel, Oct 21 2019 *)
CoefficientList[Series[1/((1-x-x^2)*(1-x^2)),{x,0,40}],x] (* Harvey P. Dale, Sep 12 2020 *)

PARI
```
{a(n) = fibonacci(n+2) - n%2};
```

[fibonacci(n+2) -(1-(-1)^n)/2 for n in (0..40)] # G. C. Greubel, Jul 10 2019

G.f.: 1/((1-x-x^2)*(1-x^2)).
a(n) = A074331(n+1).
a(n) = A054450(n+1, 1) (second column of triangle).
a(n) = 2*a(n-2) + a(n-3) + 1, with a(0)=1, a(1)=1, a(2)=3.
a(n) = Sum_{alpha=RootOf(-1+z+z^2)} (3+alpha)*alpha^(-1-n)/3 - Sum_{beta=RootOf(-1+z^2)} beta^(-1-n)/2.
a(2*k) = Sum_{j=0..k} F(2*j+1) = F(2*(k+1)) for k >= 0; a(2*k-1) = Sum_{j=0..k} F(2*j) = F(2*k+1)-1 for k >= 1 (F = A000045, Fibonacci numbers).
a(n) = a(n-1) + a(n-2) + (1+(-1)^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1, k). - Paul Barry, Oct 23 2004
a(n) = floor(phi^(n+2) / sqrt(5)), where phi is the golden ratio: phi = (1+sqrt(5))/2. - Reinhard Zumkeller, Apr 19 2005
a(n) = Fibonacci(n+1) + a(n-2) with n>1, a(0)=a(1)=1. - Zerinvary Lajos, Mar 17 2008
a(n) = floor(Fibonacci(n+3)^2/Fibonacci(n+4)). - Gary Detlefs, Nov 29 2010
a(n) = (A001595(n+3) - A066983(n+4))/2. - Gary Detlefs, Dec 19 2010
a(4*n) = F(4*n+2); a(4*n+1) = F(4*n+3) - 1; a(4*n+2) = F(4*n+4); a(4*n+3) = F(4*n+5) - 1. - Johannes W. Meijer, Jul 20 2011
a(n+1) = a(n) + a(n-1) + A059841(n+1). - Reinhard Zumkeller, Jan 06 2012
a(n) = floor(|F((1+i)*(n+2))|), n >= 0, with the complex Fibonacci function F: C -> C, z -> F(z) with F(z) := (exp(log(phi)*z) - exp(i*Pi*z)*exp(-log(phi)*z))/(2*phi-1) with the modulus |z|, the imaginary unit i and the golden section phi:=(1+sqrt(5))/2. A Conjecture: For F(z) see, e.g., the T. Koshy reference. ch. 45, p. 523, where F is called f, given in A000045. - Wolfdieter Lang, Jul 24 2012
5*a(n) = (L(n+3)-1)*(L(n+4)+3) -14 -Sum_{k=0..n} L(k+1)*L(k+5) = (L(n+3)-1)*(L(n+4)+3) -L(2*n+7) +A168309(n), where L=A000032. - J. M. Bergot, Jun 13 2014
a(n) = floor(phi*Fibonacci(n+1)), where phi is the golden section. - Michel Dekking, Dec 02 2016
a(n) = -(-1)^n * a(-4-n) for all n in Z. - Michael Somos, Dec 03 2016
a(n) = Sum_{k=0..n} Sum_{i=0..n} C(n-k-1,k-i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = floor(1/(Sum_{k>=n+4} 1/Fibonacci(k))) [Ohtsuka and Nakamura]. - Michel Marcus, Aug 09 2018
a(n) = floor(abs(chebyshevU(n/2, 3/2))). - Federico Provvedi, Feb 23 2022
E.g.f.: exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5 - sinh(x). - Stefano Spezia, Mar 09 2024

Additional formulas and more terms from Wolfdieter Lang, May 02 2000

Better description from Olivier Gérard, Jun 05 2001

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

A034867 Triangle of odd-numbered terms in rows of Pascal's triangle.

Original entry on oeis.org

1, 2, 3, 1, 4, 4, 5, 10, 1, 6, 20, 6, 7, 35, 21, 1, 8, 56, 56, 8, 9, 84, 126, 36, 1, 10, 120, 252, 120, 10, 11, 165, 462, 330, 55, 1, 12, 220, 792, 792, 220, 12, 13, 286, 1287, 1716, 715, 78, 1, 14, 364, 2002, 3432, 2002, 364, 14, 15, 455, 3003, 6435, 5005, 1365, 105, 1

Offset: 0

N. J. A. Sloane

Also triangle of numbers of n-sequences of 0,1 with k subsequences of consecutive 01 because this number is C(n+1,2*k+1). - Roger Cuculiere (cuculier(AT)imaginet.fr), Nov 16 2002
From Gary W. Adamson, Oct 17 2008: (Start)
Received from Herb Conn:
Let T = tan x, then
  tan x = T
  tan 2x = 2T / (1 - T^2)
  tan 3x = (3T - T^3) / (1 - 3T^2)
  tan 4x = (4T - 4T^3) / (1 - 6T^2 + T^4)
  tan 5x = (5T - 10T^3 + T^5) / (1 - 10T^2 + 5T^4)
  tan 6x = (6T - 20T^3 + 6T^5) / (1 - 15T^2 + 15T^4 - T^6)
  tan 7x = (7T - 35T^3 + 21T^5 - T^7) / (1 - 21T^2 + 35T^4 - 7T^6)
  tan 8x = (8T - 56T^3 + 56T^5 - 8T^7) / (1 - 28T^2 + 70T^4 - 28T^6 + T^8)
  tan 9x = (9T - 84T^3 + 126T^5 - 36T^7 + T^9) / (1 - 36 T^2 + 126T^4 - 84T^6 + 9T^8)
... To get the next one in the series, (tan 10x), for the numerator add:
  9....84....126....36....1 previous numerator +
  1....36....126....84....9 previous denominator =
  10..120....252...120...10 = new numerator
For the denominator add:
  ......9.....84...126...36...1 = previous numerator +
  1....36....126....84....9.... = previous denominator =
  1....45....210...210...45...1 = new denominator
...where numerators = A034867, denominators = A034839
(End)
Column k is the sum of columns 2k and 2k+1 of A007318. - Philippe Deléham, Nov 12 2008
Triangle, with zeros omitted, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
The row polynomials N(n,x) = Sum_{k=0..floor((n-1)/2)} T(n-1,k)*x^k, and D(n,x) = Sum_{k=0..floor(n/2)} A034839(n,k)*x^k, n >= 1, satisfy the recurrences N(n,x) = D(n-1,x) + N(n-1,x), D(n,x) = D(n-1,x) + x*N(n-1,x), with inputs N(1,x) = 1 = D(1,x). This is due to the Pascal triangle A007318 recurrence. Q(n,x) := tan(n*x)/tan(x) satisfies the recurrence Q(n,x) = (1 + Q(n-1,x))/(1 - v(x)*Q(n-1,x)) with input Q(1,x) = 1 and v = v(x) := (tan(x))^2. This recurrence is obtained from the addition theorem for tan(n*x) using n = 1 + (n-1). Therefore Q(n,x) = N(n,-v(x))/D(n,-v(x)). This proves the Gary W. Adamson contribution from above. See also A220673. This calculation was motivated by an e-mail of Thomas Olsen. The Oliver/Prodinger and Ma references resort to HAKEM Al Memo 239, Item 16, for the tan(n*x) formula in terms of tan(x). - Wolfdieter Lang, Jan 17 2013
The infinitesimal generator (infinigen) for the Narayana polynomials A090181/A001263 can be formed from the row polynomials P(n,y) of this entry. The resulting matrix is an instance of a matrix representation of the analytic infinigens presented in A145271 for general sets of binomial Sheffer polynomials and in A001263 and A119900 specifically for the Narayana polynomials. Given the column vector of row polynomials V = (1, P(1,x) = 2x, P(2,y) = 3x + x^2, P(3,y) = 4x + 4x^2, ...), form the lower triangular matrix M(n,k) = V(n-k,n-k), i.e., diagonally multiply the matrix with all ones on the diagonal and below by the components of V. Form the matrix MD by multiplying A132440^Transpose = A218272 = D (representing derivation of o.g.f.s) by M, i.e., MD = M*D. The non-vanishing component of the first row of (MD)^n * V / (n+1)! is the n-th Narayana polynomial. - Tom Copeland, Dec 09 2015
The diagonals of this entry are A078812 (also shifted A128908 and unsigned A053122, which are embedded in A030528, A102426, A098925, A109466, A092865). Equivalently, the antidiagonals of A078812 are the rows of A034867. - Tom Copeland, Dec 12 2015
Binomial(n,2k+1) is also the number of permutations avoiding both 132 and 213 with k peaks, i.e., positions with w[i]w[i+2]. - Lara Pudwell, Dec 19 2018
Binomial(n,2k+1) is also the number of permutations avoiding both 123 and 132 with k peaks, i.e., positions with w[i]w[i+2]. - Lara Pudwell, Dec 19 2018
The row polynomial P(n, x) = Sum_{0..floor(n/2)} T(n, k)*x^k appears as numerator polynomial of the diagonal sequence m of triangle A104698 as follows. G(m, x) = P(m, x^2)/(1 - x)^(m+1), for m >= 0. - Wolfdieter Lang, May 14 2025
Number of acyclic orientations of the path graph on n+1 vertices, with k-1 sinks. - Per W. Alexandersson, Aug 15 2025

			Triangle T starts:
  n\k   0   1   2   3   4  5 ...   ----------------------------------------
0:    1
1:    2
2:    3   1
3:    4   4
4:    5  10   1
5:    6  20   6
6:    7  35  21   1
7:    8  56  56   8
8:    9  84 126  36   1
9:   10 120 252 120  10
 10:   11 165 462 330  55  1
 11:   12 220 792 792 220 12
... ... reformatted and extended by - _Wolfdieter Lang_, May 14 2025

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 136.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.
Sergi Elizalde, Johnny Rivera Jr., and Yan Zhuang, Counting pattern-avoiding permutations by big descents, arXiv:2408.15111 [math.CO], 2024. See p. 6.
S.-M. Ma, On some binomial coefficients related to the evaluation of tan(nx), arXiv preprint arXiv:1205.0735 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 13 2012
K. Oliver and H. Prodinger, The continued fraction expansion of Gauss' hypergeometric function and a new application to the tangent function, Transactions of the Royal Society of South Africa, Vol. 76 (2012), 151-154, [DOI], [PDF]. - From _N. J. A. Sloane_, Jan 03 2013
Eric Weisstein's World of Mathematics, Tangent. [From _Eric W. Weisstein_, Oct 18 2008]

Cf. A000129, A007318, A034839, A034867, A084938, A131980, A220673.
Cf. A001263, A090181, A119900, A132440, A145271.
Cf. A030528, A053122, A078812, A092865, A098925, A102426, A109466, A128908, A218272, A104698.
From Wolfdieter Lang, May 14 2025:(Start)
Row length A008619. Row sums A000079. Alternating row sums  A009545(n+1).
Column sequences (with certain offsets): A000027, A000292, A000389, A000580, A000582, A001288, ... (End)

/* as a triangle */ [[Binomial(n+1,2*k+1): k in [0..Floor(n/2)]]: n in [0..20]]; // G. C. Greubel, Mar 06 2018

seq(seq(binomial(n+1,2*k+1), k=0..floor(n/2)), n=0..14); # Emeric Deutsch, Apr 01 2005

u[1, x_] := 1; v[1, x_] := 1; z = 12;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]  (* A034839 as a triangle *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]  (* A034867 as a triangle *)
(* Clark Kimberling, Feb 18 2012 *)
Table[Binomial[n+1, 2*k+1], {n,0,20}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Mar 06 2018 *)

for(n=0,20, for(k=0,floor(n/2), print1(binomial(n+1,2*k+1), ", "))) \\ G. C. Greubel, Mar 06 2018

T(n,k) = C(n+1,2k+1) = Sum_{i=k..n-k} C(i,k) * C(n-i,k).
E.g.f.: 1+(exp(x)*sinh(x*sqrt(y)))/sqrt(y). - Vladeta Jovovic, Mar 20 2005
G.f.: 1/((1-z)^2-t*z^2). - Emeric Deutsch, Apr 01 2005
T(n,k) = Sum_{j = 0..n} A034839(j,k). - Philippe Deléham, May 18 2005
Pell(n+1) = A000129(n+1) = Sum_{k=0..n} T(n,k) * 2^k = (1/n!) Sum_{k=0..n} A131980(n,k) * 2^k. - Tom Copeland, Nov 30 2007
T(n,k) = A007318(n,2k) + A007318(n,2k+1). - Philippe Deléham, Nov 12 2008
O.g.f for column k, k>=0: (1/(1-x)^2)*(x/(1-x))^(2*k). See the G.f. of this array given above by Emeric Deutsch. - Wolfdieter Lang, Jan 18 2013
T(n,k) = (x^(2*k+1))*((1+x)^n-(1-x)^n)/2. - L. Edson Jeffery, Jan 15 2014

More terms from Emeric Deutsch, Apr 01 2005

A125145 a(n) = 3a(n-1) + 3a(n-2). a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 15, 57, 216, 819, 3105, 11772, 44631, 169209, 641520, 2432187, 9221121, 34959924, 132543135, 502509177, 1905156936, 7222998339, 27384465825, 103822392492, 393620574951, 1492328902329, 5657848431840, 21450532002507

Offset: 0

Tanya Khovanova, Jan 11 2007

Number of aa-avoiding words of length n on the alphabet {a,b,c,d}.
Equals row 3 of the array shown in A180165, the INVERT transform of A028859 and the INVERTi transform of A086347. - Gary W. Adamson, Aug 14 2010
From Tom Copeland, Nov 08 2014: (Start)
This array is one of a family related by compositions of C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for A000108; its inverse Cinv(x) = x(1-x); and the special Mobius transformation P(x,t) = x / (1+t*x) with inverse P(x,-t) in x. Cf. A091867.
O.g.f.: G(x) = P[P[P[-Cinv(-x),-1],-1],-1] = P[-Cinv(-x),-3] = x*(1+x)/[1-3x(1-x)]= x*A125145(x).
Ginv(x) = -C[-P(x,3)] = [-1 + sqrt(1+4x/(1+3x))]/2 = x*A104455(-x).
G(-x) = -x(1-x) * [ 1 - 3*[x*(1+x)] + 3^2*[x*(1+x)]^2 - ...] , and so this array is related to finite differences in the row sums of A030528 * Diag((-3)^1,3^2,(-3)^3,..). (Cf. A146559.)
The inverse of -G(-x) is C[-P(-x,3)]= [1 - sqrt(1-4x/(1-3x))]/2 = x*A104455(x). (End)
Number of 3-compositions of n+1 restricted to parts 1 and 2 (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
Joerg Arndt, Matters Computational (The Fxtbook)
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 7.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,3).

Cf. A028859 = a(n+2) = 2 a(n+1) + 2 a(n); A086347 = On a 3 X 3 board, number of n-move routes of chess king ending at a given side cell. a(n) = 4a(n-1) + 4a(n-2).
Cf. A128235.
Cf. A180165, A028859, A086347. - Gary W. Adamson, Aug 14 2010
Cf. A002605, A026150, A030195, A080040, A083337, A106435, A108898.
Cf. A000108, A091867, A125145, A104455, A030528, A146559.

a125145 n = a125145_list !! n
a125145_list =
   1 : 4 : map (* 3) (zipWith (+) a125145_list (tail a125145_list))
-- Reinhard Zumkeller, Oct 15 2011

I:=[1,4]; [n le 2 select I[n] else 3*Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 10 2014

a[0]:=1: a[1]:=4: for n from 2 to 27 do a[n]:=3*a[n-1]+3*a[n-2] od: seq(a[n],n=0..27); # Emeric Deutsch, Feb 27 2007
A125145 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[1,4]) ;
    else
        3*(procname(n-1)+procname(n-2)) ;
    end if;
end proc: # R. J. Mathar, Feb 13 2022

nn=23;CoefficientList[Series[(1+x)/(1-3x-3x^2),{x,0,nn}],x] (* Geoffrey Critzer, Feb 09 2014 *)
LinearRecurrence[{3,3},{1,4},30] (* Harvey P. Dale, May 01 2022 *)

G.f.: (1+z)/(1-3z-3z^2). - Emeric Deutsch, Feb 27 2007
a(n) = (5*sqrt(21)/42 + 1/2)*(3/2 + sqrt(21)/2)^n + (-5*sqrt(21)/42 + 1/2)*(3/2 - sqrt(21)/2)^n. - Antonio Alberto Olivares, Mar 20 2008
a(n) = A030195(n)+A030195(n+1). - R. J. Mathar, Feb 13 2022
E.g.f.: exp(3*x/2)*(21*cosh(sqrt(21)*x/2) + 5*sqrt(21)*sinh(sqrt(21)*x/2))/21. - Stefano Spezia, Aug 04 2022
a(n) = (((3 + sqrt(21)) / 2)^(n+2) - ((3 - sqrt(21)) / 2)^(n+2)) / (3 * sqrt(21)). - Werner Schulte, Dec 17 2024

A115139 Array of coefficients of polynomials related to integer powers of the generating function of Catalan numbers A000108.

Original entry on oeis.org

1, 1, 1, -1, 1, -2, 1, -3, 1, 1, -4, 3, 1, -5, 6, -1, 1, -6, 10, -4, 1, -7, 15, -10, 1, 1, -8, 21, -20, 5, 1, -9, 28, -35, 15, -1, 1, -10, 36, -56, 35, -6, 1, -11, 45, -84, 70, -21, 1, 1, -12, 55, -120, 126, -56, 7, 1, -13, 66, -165, 210, -126, 28, -1, 1, -14, 78, -220, 330, -252, 84, -8, 1, -15, 91, -286, 495, -462, 210, -36, 1

Offset: 1

Wolfdieter Lang, Jan 13 2006

This is a signed version of A011973 (Fibonacci polynomials) with different offset.
The sequence of row lengths is [1,1,2,2,3,3,4,4,5,5,6,6,...] = A008619(n-1), n>=1.
The row sums give the period 6 sequence [1,1,0,-1,-1,0,...] = A010892(n-1), n>=1.
The o.g.f. for the column m sequence (with leading zeros) is ((-1)^m)*x^(2*m+1)/(1-x)^(m+1).
The unsigned row sums give the Fibonacci numbers A000045(n-1), n>=1.
The row polynomial are P(n,x):= Sum_{m=0..ceiling(n/2)-1} a(n,m)*x^m = (sqrt(x)^(n-1))*S(n-1,1/sqrt(x)), n>=1, with Chebyshev's S(n,x) polynomials A049310.
These polynomials appear in the formula 1/c(x)^n = P(n+1,x) - x*P(n,x)*c(x), n>=1, with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers). See the W. Lang reference, eqs. (1) and (2), p. 408, with P(n,x):=p(-n,x).
These polynomials also appear in the formula c(x)^n = (-P(n-1,x) + P(n,x)*c(x))/x^(n-1), n>=1, with the above given o.g.f. c(x) of A000108 (Catalan numbers). See the W. Lang reference, eq. (1), with P(n,x):=p(-n,x).
With offset n>=0 this array a(n,m) coincides with the row reversed coefficient table of Chebyshev's S-polynomials without interspersed zeros. See A049310 for the S(n,x) coefficient table with increasing powers of x.
The polynomials with this sequence as coefficients form the set of so-called "Catalan polynomials", having arisen from computations in looking at the problem of 'fitting' iterated generating function schemes to the Catalan sequence. A neighboring pair forms the basis of a first-order linear recurrence that generates, through a succession of iterated generating functions (polynomials in Z[x]), a predetermined number of Catalan numbers before 'failing' - see the Clapperton et al. 2008 reference in Utilitas Mathematica, where some of the essential mathematical properties of the Catalan polynomials are also listed (based mainly on existing results for Dickson and Chebyshev polynomials, to which they are related). - Peter J Larcombe, Sep 16 2008
In the Clapperton et al. 2008 Congressus Numerantium paper, a new class of nonlinear identities satisfied by Catalan polynomials are presented. They arise from the algebraic implementation of particular cases of a general root finding formulation due to Householder, of which the classic O(2) Newton-Raphson and O(3) Halley algorithms are special cases. The role of Catalan polynomials in forming Padé approximants to the Catalan sequence o.g.f. is also discussed. - Peter J Larcombe, Nov 02 2008
These polynomials appear in the following statements: (i) P(k+1,x)/P(k+2,x) is the g.f. of all ordered trees (Dyck paths) of height at most k; (ii) x^k/(P(k+1,x)*P(k+2,x)) is the g.f. of all ordered trees (Dyck paths) of height k. See the de Bruijn et al., the Kreweras, the Sedgewick and Flajolet (p. 258), and the Flajolet and Sedgewick (p. 326) references. - Emeric Deutsch, Jun 16 2011
For a mirrored, shifted version showing the relation of these coefficients to the Pascal triangle, Fibonacci, and other number triangles, see A030528. See also A053122 for a relation to Cartan matrices. (Cf. A011973, A169803, A115139, A092865, A098925, and A102426.) - Tom Copeland, Nov 04 2014
M. Sinan Kul, Dec 09 2015, observed that (in a rewritten form) Chebyshev's S polynomials A049310 are given by S(n, x) = Sum_{m=0..floor(n/2)} a(n+1, m)*x^(n-2*m), n >= 0. This formula is well known and can be proved from the S recurrence by induction using the recurrence for the binomial coefficients. - Wolfdieter Lang, Feb 01 2016
These are the coefficients of generalized Fibonacci polynomials (see link bellow). - Rigoberto Florez, Aug 28 2022

			The irregular triangle a(n, m) begins:
  n\m  0   1   2    3   4    5   6   7  8
  1:   1
  2:   1
  3:   1  -1
  4:   1  -2
  5:   1  -3   1
  6:   1  -4   3
  7:   1  -5   6   -1
  8:   1  -6  10   -4
  9:   1  -7  15  -10   1
  10:  1  -8  21  -20   5
  11:  1  -9  28  -35  15   -1
  12:  1 -10  36  -56  35   -6
  13:  1 -11  45  -84  70  -21   1
  14:  1 -12  55 -120 126  -56   7
  15:  1 -13  66 -165 210 -126  28  -1
  16:  1 -14  78 -220 330 -252  84  -8
  17:  1 -15  91 -286 495 -462 210 -36  1
  ... Reformatted and extended. - _Wolfdieter Lang_, Jan 27 2016
1/c(x) = P(2,x) - x*P(1,x)*c(x) = 1 - x*c(x), with the o.g.f. of A000108 (Catalan).
1/c(x)^2 = P(3,x) - x*P(2,x)*c(x) = (1-x) - x*c(x).
c(x)^2 = (-P(1,x) + P(2,x)*c(x))/x^1 = (-1 + 1*c(x))/x.
c(x)^3 = (-P(2,x) + P(3,x)*c(x))/x^2 = (-1 + (1-x)*c(x))/x^2.
P(3,x) = 1-x = x*S(2,1/sqrt(x)) with Chebyshev's S(2,y) = U(2,y/2) = y^2 - 1.

J. A. Clapperton, P. J. Larcombe and E. J. Fennessey, On iterated generating functions for integer sequences and Catalan polynomials, Utilitas Mathematica, 77 (2008), 3-33.
J. A. Clapperton, P. J. Larcombe, E. J. Fennessey and P. Levrie, A class of auto-identities for Catalan polynomials and Padé approximation, Congressus Numerantium, 189 (2008), 77-95.
N. G. de Bruijn, D. E. Knuth and S. O. Rice, The average height of planted plane trees, in: Graph Theory and Computing (ed. T. C. Read), Academic Press, New York, 1972, pp. 15-22.
R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.

Alois P. Heinz, Rows n = 1..200, flattened
Hong Duc Bui, Counting Number of Triangulations of Point Sets: Reinterpreting and Generalizing the Triangulation Polynomials, arXiv:2504.09615 [cs.CG], 2025. See p. 3.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009.
R. Florez and J. C. Saunders, Irreducibility of generalized Fibonacci polynomials , Integers 22 (2022).
Atsushi Komaba, Hisashi Johno, and Kazunori Nakamoto, A novel statistical approach for two-sample testing based on the overlap coefficient, arXiv:2206.03166 [math.ST], 2022.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
Wolfdieter Lang, First 16 rows
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419.
Peter J. Larcombe and Eric J. Fennessey, A Non-Linear Identity for A Particular Class of Polynomial Families, Fibonacci Quart. 52 (2014), no. 1, 75-79. Mentions this sequence.

Cf. A011973, A030528, A053122, A092865, A098925, A102426, A115139, A169803.

seq(seq((-1)^k*binomial(n-k,k),k=0..floor(n/2)),n=0..16); # Peter Luschny, May 10 2016

p[x_, n_] := p[x, n] = p[x, n - 1] + x*p[x, n - 2];
p[x_, -1] = p[x_, 0] = 1; p[x_, 1] = 1 + x;
Flatten[ Table[ CoefficientList[p[-x, n - 1], x], {n, 0, 16}]]
(* Jean-François Alcover, Jun 20 2011 *)
Flatten[Map[CoefficientList[#,x]&, Table[Sum[Binomial[t - i, i] x^(i) (-1)^i, {i, 0, t}], {t, 1,15}]]] (* Rigoberto Florez, Aug 28 2022 *)

import math
L1 = [math.comb(t - i, i)*(-1)**i for t in range(16) for i in range(t)]
L1 = list(filter((0)._ne_, L1))
print(L1) # Rigoberto Florez, Sep 03 2022

a(n, m) = ((-1)^(m))*binomial(n-1-m, m), n>=1, m=0..ceiling(n/2)-1.
a(n, m) = [x^m]P(n,x), n>=1, m=0..ceiling(n/2)-1, with P(n,x) given above in terms of Chebyshev's S-polynomials.
P(n,x) = (u^(2*n) - v^(2*n))/(u^2 - v^2), where u and v are defined by u^2 + v^2 =1 and u*v = sqrt(x). Example: P(3,x) = (u^6 - v^6)/(u^2 - v^2) = u^4 + u^2*v^2 + v^4 = 1 - x. - Emeric Deutsch, Jun 16 2011
G.f.: 1/(1- x + y*x^2) = R(0)/2, where R(k) = 1 + 1/(1 - (2*k+1- x*y)*x/((2*k+2- x*y)*x + 1/R(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 09 2013
T(n, k) = GegenbauerC(k, (n+1)/2-k, -1) assuming the triangle (0,0) based. - Peter Luschny, May 10 2016

A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j+1).

Original entry on oeis.org

0, 1, 2, 3, 3, 0, -9, -27, -54, -81, -81, 0, 243, 729, 1458, 2187, 2187, 0, -6561, -19683, -39366, -59049, -59049, 0, 177147, 531441, 1062882, 1594323, 1594323, 0, -4782969, -14348907, -28697814, -43046721, -43046721, 0, 129140163, 387420489, 774840978

Offset: 0

N. J. A. Sloane, Oct 20 2000

Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1-M)^n = A057681(n)-a(n)*M+z(n)*M^2, where z(0)=z(1)=0 and, apparently, z(n+2)=A057083(n). - Stanislav Sykora, Jun 10 2012
From Tom Copeland, Nov 09 2014: (Start)
This array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interp. (here t=-2) o.g.f. G(x,t) = x(1-x)/[1+(t-1)x(1-x)] and inverse o.g.f. Ginv(x,t) =  [1-sqrt(1-4x/(1+(1-t)x))]/2 (Cf. A005773 and A091867 and A030528 for more info on this family). (End)
{A057681, A057682, A*}, where A* is A057083 prefixed by two 0's, is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x)} of order 3. For the definitions of {k_i(x)} and the difference analog {K_i (n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Jul 31 2017

			G.f. = x + 2*x^2 + 3*x^3 + 3*x^4 - 9*x^6 - 27*x^7 - 54*x^8 - 81*x^9 + ...
If M^3=1 then (1-M)^6 = A057681(6) - a(6)*M + A057083(4)*M^2 = -18 + 9*M + 9*M^2. - _Stanislav Sykora_, Jun 10 2012

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mark W. Coffey, Reductions of particular hypergeometric functions 3F2 (a, a+1/3, a+2/3; p/3, q/3; +-1), arXiv preprint arXiv:1506.09160 [math.CA], 2015.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3).

Alternating row sums of triangle A030523.

Cf. A000108, A000484, A000748, A005043, A005773, A057083, A057681, A091867.

I:=[0,1,2]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

A057682:=n->add((-1)^j*binomial(n,3*j+1), j=0..floor(n/3)):
seq(A057682(n), n=0..50); # Wesley Ivan Hurt, Nov 11 2014

A[n_] := Array[KroneckerDelta[#1, #2 + 1] - KroneckerDelta[#1, #2] + Sum[KroneckerDelta[#1, #2 -q], {q, n}] &, {n, n}];
Join[{0,1}, Table[(-1)^(n-1)*Total[CoefficientList[ CharacteristicPolynomial[A[(n-1)], x], x]], {n,2,30}]] (* John M. Campbell, Mar 16 2012 *)
Join[{0}, LinearRecurrence[{3,-3}, {1,2}, 40]] (* Jean-François Alcover, Jan 08 2019 *)

{a(n) = sum( j=0, n\3, (-1)^j * binomial(n, 3*j + 1))} /* Michael Somos, May 26 2004 */

{a(n) = if( n<2, n>0, n-=2; polsym(x^2 - 3*x + 3, n)[n + 1])} /* Michael Somos, May 26 2004 */

b=BinaryRecurrenceSequence(3,-3,1,2)
def A057682(n): return 0 if n==0 else b(n-1)
[A057682(n) for n in range(41)] # G. C. Greubel, Jul 14 2023

G.f.: (x - x^2) / (1 - 3*x + 3*x^2).
a(n) = 3*a(n-1) - 3*a(n-2), if n>1.
Starting at 1, the binomial transform of A000484. - Paul Barry, Jul 21 2003
It appears that abs(a(n)) = floor(abs(A000748(n))/3). - John W. Layman, Sep 05 2003
a(n) = ((3+i*sqrt(3))/2)^(n-2) + ((3-i*sqrt(3))/2)^(n-2). - Benoit Cloitre, Oct 27 2003
a(n) = n*3F2(1/3-n/3,2/3-n/3,1-n/3 ; 2/3,4/3 ; 1) for n>=1. - John M. Campbell, Jun 01 2011
Let A(n) be the n X n matrix with -1's along the main diagonal, 1's everywhere above the main diagonal, and 1's along the subdiagonal. Then a(n) equals (-1)^(n-1) times the sum of the coefficients of the characteristic polynomial of A(n-1), for all n>1 (see Mathematica code below). - John M. Campbell, Mar 16 2012
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) - z(n), y(n+1) = y(n) - x(n), z(n+1) = z(n) - y(n). Then a(n) = -y(n). But this recurrence falls into a repetitive cycle of length 6 and multiplicative factor -27, so that a(n) = -27*a(n-6) for any n>6. - Stanislav Sykora, Jun 10 2012
a(n) = A057083(n-1) - A057083(n-2). - R. J. Mathar, Oct 25 2012
G.f.: 3*x - 1/3 + 3*x/(G(0) - 1) where G(k)= 1 + 3*(2*k+3)*x/(2*k+1 - 3*x*(k+2)*(2*k+1)/(3*x*(k+2) + (k+1)/G(k+1)));(continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
G.f.: Q(0,u) -1, where u=x/(1-x), Q(k,u) = 1 - u^2 + (k+2)*u - u*(k+1 - u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
From Vladimir Shevelev, Jul 31 2017: (Start)
For n>=1, a(n) = 2*3^((n-2)/2)*cos(Pi*(n-2)/6);
For n>=2, a(n) = K_1(n) + K_3(n-2);
For m,n>=2, a(n+m) = a(n)*K_1(m) + K_1(n)*a(m) - K_3(n-2)*K_3(m-2), where
K_1 = A057681, K_3 = A057083. (End)

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

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A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

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A146559 Expansion of (1-x)/(1 - 2*x + 2*x^2).

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A026729 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.

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A052952 a(n) = Fibonacci(n+2) - (1-(-1)^n)/2.

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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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A034867 Triangle of odd-numbered terms in rows of Pascal's triangle.

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A146559 Expansion of (1-x)/(1 - 2x + 2x^2).

A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j+1).

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