A002212 -id:A002212 - cp's OEIS frontend

A130777 Coefficients of first difference of Chebyshev S polynomials.

1, -1, 1, -1, -1, 1, 1, -2, -1, 1, 1, 2, -3, -1, 1, -1, 3, 3, -4, -1, 1, -1, -3, 6, 4, -5, -1, 1, 1, -4, -6, 10, 5, -6, -1, 1, 1, 4, -10, -10, 15, 6, -7, -1, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, -1, -5, 15, 20, -35, -21, 28, 8, -9, -1, 1, 1, -6, -15, 35, 35, -56, -28, 36, 9, -10, -1, 1

Offset: 0

Philippe Deléham, Jul 14 2007

Inverse of triangle in A061554.
Signed version of A046854.
From Paul Barry, May 21 2009: (Start)
Riordan array ((1-x)/(1+x^2),x/(1+x^2)).
This triangle is the coefficient triangle for the Hankel transforms of the family of generalized Catalan numbers that satisfy a(n;r)=r*a(n-1;r)+sum{k=1..n-2, a(k)*a(n-1-k;r)}, a(0;r)=a(1;r)=1. The Hankel transform of a(n;r) is h(n)=sum{k=0..n, T(n,k)*r^k} with g.f. (1-x)/(1-r*x+x^2). These sequences include A086246, A000108, A002212. (End)
From Wolfdieter Lang, Jun 11 2011: (Start)
The Riordan array ((1+x)/(1+x^2),x/(1+x^2)) with entries Phat(n,k)= ((-1)^(n-k))*T(n,k) and o.g.f. Phat(x,z)=(1+z)/(1-x*z+z^2) for the row polynomials Phat(n,x) is related to Chebyshev C and S polynomials as follows.
Phat(n,x) = (R(n+1,x)-R(n,x))/(x+2) = S(2*n,sqrt(2+x))
  with R(n,x)=C_n(x) in the Abramowitz and Stegun notation, p. 778, 22.5.11. See A049310 for the S polynomials. Proof from the o.g.f.s.
Recurrence for the row polynomials Phat(n,x):
  Phat(n,x) = x*Phat(n-1,x) - Phat(n-2,x) for n>=1; Phat(-1,x)=-1, Phat(0,x)=1.
The A-sequence for this Riordan array Phat (see the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices) is given by 1, 0, -1, 0, -1, 0, -2, 0, -5,.., starting with 1 and interlacing the negated A000108 with zeros (o.g.f. 1/c(x^2) = 1-c(x^2)*x^2, with the o.g.f. c(x) of A000108).
The Z-sequence has o.g.f. sqrt((1-2*x)/(1+2*x)), and it is given by A063886(n)*(-1)^n.
The A-sequence of the Riordan array T(n,k) is identical with the one for the Riordan array Phat, and the Z-sequence is -A063886(n).
(End)
The row polynomials P(n,x) are the characteristic polynomials of the adjacency matrices of the graphs which look like P_n (n vertices (nodes), n-1 lines (edges)), but vertex no. 1 has a loop. - Wolfdieter Lang, Nov 17 2011
From Wolfdieter Lang, Dec 14 2013: (Start)
The zeros of P(n,x) are x(n,j) = -2*cos(2*Pi*j/(2*n+1)), j=1..n. From P(n,x) = (-1)^n*S(2*n,sqrt(2-x)) (see, e.g., the Lemma 6 of the W. Lang link).
The discriminants of the P-polynomials are given in A052750. (End)

			The triangle T(n,k) begins:
n\k  0   1   1   3    4    5    6    7    8    9  10  11  12  13 14 15 ...
0:   1
1:  -1   1
2:  -1  -1   1
3:   1  -2  -1   1
4:   1   2  -3  -1    1
5:  -1   3   3  -4   -1    1
6:  -1  -3   6   4   -5   -1    1
7:   1  -4  -6  10    5   -6   -1    1
8:   1   4 -10 -10   15    6   -7   -1    1
9:  -1   5  10 -20  -15   21    7   -8   -1    1
10: -1  -5  15  20  -35  -21   28    8   -9   -1   1
11:  1  -6 -15  35   35  -56  -28   36    9  -10  -1   1
12:  1   6 -21 -35   70   56  -84  -36   45   10 -11  -1   1
13: -1   7  21 -56  -70  126   84 -120  -45   55  11 -12  -1   1
14: -1  -7  28  56 -126 -126  210  120 -165  -55  66  12 -13  -1  1
15:  1  -8 -28  84  126 -252 -210  330  165 -220 -66  78  13 -14 -1  1
...  reformatted and extended - _Wolfdieter Lang_, Jul 31 2014.
---------------------------------------------------------------------------
From _Paul Barry_, May 21 2009: (Start)
Production matrix is
-1, 1,
-2, 0, 1,
-2, -1, 0, 1,
-4, 0, -1, 0, 1,
-6, -1, 0, -1, 0, 1,
-12, 0, -1, 0, -1, 0, 1,
-20, -2, 0, -1, 0, -1, 0, 1,
-40, 0, -2, 0, -1, 0, -1, 0, 1,
-70, -5, 0, -2, 0, -1, 0, -1, 0, 1 (End)
Row polynomials as first difference of S polynomials:
P(3,x) = S(3,x) - S(2,x) = (x^3 - 2*x) - (x^2 -1) = 1 - 2*x - x^2 +x^3.
Alternative triangle recurrence (see a comment above): T(6,2) = T(5,2) + T(5,1) = 3 + 3 = 6. T(6,3) = -T(5,3) + 0*T(5,1) = -(-4) = 4. - _Wolfdieter Lang_, Jul 31 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available).

Hyeong-Kwan Ju, On the sequence generated by a certain type of matrices, Honam Math. J. 39, No. 4, 665-675 (2017), Theorem 2.16.
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012-2017; see Definition 1, Lemma 6 and Remark 4.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A066170, A046854, A057077 (first column).

Row sums: A010892(n+1); repeat(1,0,-1,-1,0,1). Alternating row sums: A061347(n+2); repeat(1,-2,1).

A130777 := proc(n,k): (-1)^binomial(n-k+1,2)*binomial(floor((n+k)/2),k) end: seq(seq(A130777(n,k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

T[n_, k_] := (-1)^Binomial[n - k + 1, 2]*Binomial[Floor[(n + k)/2], k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2017, from Maple *)

@CachedFunction
def A130777(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = A130777(n-1,k) if n==1 else 0
    return A130777(n-1,k-1) - A130777(n-2,k) - h
for n in (0..9): [A130777(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Number triangle T(n,k) = (-1)^C(n-k+1,2)*C(floor((n+k)/2),k). - Paul Barry, May 21 2009
From Wolfdieter Lang, Jun 11 2011: (Start)
Row polynomials: P(n,x) = sum(k=0..n, T(n,k)*x^k) = R(2*n+1,sqrt(2+x)) / sqrt(2+x), with Chebyshev polynomials R with coefficients given in A127672 (scaled T-polynomials).
R(n,x) is called C_n(x) in Abramowitz and Stegun's handbook, p. 778, 22.5.11.
P(n,x) = S(n,x)-S(n-1,x), n>=0, S(-1,x)=0, with the Chebyshev S-polynomials (see the coefficient triangle A049310).
O.g.f. for row polynomials: P(x,z):= sum(n>=0, P(n,x)*z^n ) = (1-z)/(1-x*z+z^2).
  (from the o.g.f. for R(2*n+1,x), n>=0, computed from the o.g.f. for the R-polynomials (2-x*z)/(1-x*z+z^2) (see A127672))
Proof of the Chebyshev connection from the o.g.f. for Riordan array property of this triangle (see the P. Barry comment above).
For the A- and Z-sequences of this Riordan array see a comment above. (End)
abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) T(n,n-k) = A108299(n,k); abs(T(n,n-k)) = A065941(n,k). - Johannes W. Meijer, Aug 08 2011
From Wolfdieter Lang, Jul 31 2014: (Start)
Similar to the triangles A157751, A244419 and A180070 one can give for the row polynomials P(n,x) besides the usual three term recurrence another one needing only one recurrence step. This uses also a negative argument, namely P(n,x) = (-1)^(n-1)*(-1 + x/2)*P(n-1,-x) + (x/2)*P(n-1,x), n >= 1, P(0,x) = 1. Proof by computing the o.g.f. and comparing with the known one. This entails the alternative triangle recurrence T(n,k) = (-1)^(n-k)*T(n-1,k) + (1/2)*(1 + (-1)^(n-k))*T(n-1,k-1), n >= m >= 1, T(n,k) = 0 if n < k and T(n,0) = (-1)^floor((n+1)/2) = A057077(n+1). [P(n,x) recurrence corrected Aug 03 2014]
(End)

New name and Chebyshev comments by Wolfdieter Lang, Jun 11 2010

A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells.

Original entry on oeis.org

0, 1, 1, 2, 5, 12, 37, 123, 446, 1689, 6693, 27034, 111630, 467262, 1981353, 8487400, 36695369, 159918120, 701957539, 3101072051, 13779935438, 61557789660, 276327463180, 1245935891922, 5640868033058, 25635351908072, 116911035023017

Offset: 0

N. J. A. Sloane

Named after the American mathematician Frank Harary (1921-2005) and the British mathematician Ronald Cedric Read (1924-2019). - Amiram Eldar, Jun 22 2021

S. J. Cyvin, J. Brunvoll, X. F. Guo and F. J. Zhang, Number of perifusenes with one internal vertex, Rev. Roumaine Chem., Vol. 38, No. 1 (1993), pp. 65-77.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem., Vol. 134, No. 1 (1997), pp. 55-70.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron, Vol. 42, No. 19 (1986), pp. 5291-5299. See Table 3.
J. V. Knop, K. Szymansky, Željko Jeričević and Nenad Trinajstić, On the total number of polyhexes, Match, Vol. 16 (1984), pp. 119-134.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer generation of isomeric structures, Pure & Appl. Chem., Vol. 55, No. 2 (1983), pp. 379-390.

T. D. Noe, Table of n, a(n) for n = 0..200
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147. [Annotated scanned copy]
S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Harary-Read numbers for catafusenes: Complete classification according to symmetry, Journal of mathematical chemistry, Vol. 9, No. 1 (1992), pp. 19-31 and 33-38. See Table 2.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc., Vol. 17, No. 1 (1970), pp. 1-13; alternative link.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
R. C. Read, Letter to N. J. A. Sloane, Feb 12 1971. (includes 40 terms of A002212 and A002216)
Eric Weisstein's World of Mathematics, Polyhex.
Eric Weisstein's World of Mathematics, Fusene.

Cf. A036359, A005963, A000228, A001998.

Cf. A002212, A002213, A002214, A002215.

CoefficientList[Series[(12+(1-5*x)^(3/2)*(1-x)^(3/2)+24*x-48*x^2- 24*x^3- 3*(3+5 x)*Sqrt[1-5*x^2]*Sqrt[1-x^2]-4*Sqrt[1-5*x^3]*Sqrt[1-x^3])/ (24*x^2),{x,0,40}],x] (* Harvey P. Dale, Dec 23 2013 *)

G.f.: (1/(24*x^2))*(12+24*x-48*x^2-24*x^3 +(1-x)^(3/2)*(1-5*x)^(3/2)-3*(3+5*x)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2) -4*(1-x^3)^(1/2)*(1-5*x^3)^(1/2)).
a(n) = (1/2)[A002214(n)+A002215(n)], n>=1. - Emeric Deutsch, Dec 23 2003
a(n) ~ 5^(n+1/2)/(4*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Aug 09 2013

A032357 Convolution of Catalan numbers and powers of -1.

Original entry on oeis.org

1, 0, 2, 3, 11, 31, 101, 328, 1102, 3760, 13036, 45750, 162262, 580638, 2093802, 7601043, 27756627, 101888163, 375750537, 1391512653, 5172607767, 19293659253, 72188904387, 270870709263, 1019033438061, 3842912963391, 14524440108761

Offset: 0

Wolfdieter Lang

Absolute value of the alternating sum of Catalan Numbers. - Alexander Adamchuk, Jul 03 2006
Sums of two consecutive terms are a(n-1) + a(n) = 1, 2, 5, 14, 42, ... = A000108(n) (Catalan Numbers). The prime p divides a((p-3)/2) for p = 11, 19, 29, 31, 41, 59, 61, 71, ... = A045468 (Primes congruent to {1, 4} mod 5). Prime p divides a(2*p+1) for p = 5, 11, 19, 29, 31, 41, 59, 61, 71, ... = A038872 (Primes congruent to {0, 1, 4} mod 5). Also odd primes where 5 is a square mod p. - Alexander Adamchuk, Jul 03 2006
Hankel transform is F(2*n+1), where F = A000045. - Paul Barry, Jul 22 2008
Equals INVERTi transform of A000958. - Gary W. Adamson, Apr 10 2009
Inverse binomial transform of A002212. - Philippe Deléham, Sep 17 2009
Number of singleton and plus-decomposable (2143, 2413, 3142)-avoiding permutations with no +bonds (ascents by 1), with offset 1. Equivalently, number of (2143, 2413, 3142)-avoiding permutations that start with 1 or end with n (top entry). E.g., 132 and 213 for n = 3; 1324, 1432, 3214 for n = 4. - Alexander Burstein, May 22 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad and Mingjia Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, 2017.

Cf. A000045, A000108, A000958, A001622, A002212, A014137, A014138, A033297, A038872, A045468, A064739.

rec:= (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-2*n+1)*a(n-2)=0:
A:= gfun:-rectoproc({rec,a(0)=1,a(1)=0},a(n),remember):
seq(A(n),n=0..50); # Robert Israel, May 22 2015

Table[Sum[(-1)^(k+n)*CatalanNumber[k],{k,0,n}],{n,0,60}] (* Alexander Adamchuk, Jul 03 2006 *)
Round@Table[(-1)^n/GoldenRatio + CatalanNumber[n + 1] Hypergeometric2F1[1, n + 3/2, n + 3, -4], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 02 2016 *)
Table[(CatalanNumber[n] (2 + (n + 1) Hypergeometric2F1[1, -n, 1/2, 5/4]) - (-1)^n)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

def A032357():
    f, c, n = 1, 1, 1
    while True:
        yield f
        n += 1
        c = c * (4*n - 6) // n
        f = c - f
a = A032357()
print([next(a) for  in range(27)]) # _Peter Luschny, Nov 30 2016

G.f.: c(x)/(1 + x), where c(x) is the g.f. for the Catalan numbers A000108.
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k), where C(k) = A000108(k).
a(n) = ((-1)^(n+1) - binomial(2*(n+1), n+1)*Sum_{k=0..n+1} (-5)^k*binomial(n+1, k)/binomial(2*k, k))/2.
a(n) = C(2*n, n)/(n+1) - a(n-1) = A000108(n) - a(n-1) with a(0) = 1. - Labos Elemer, Apr 26 2003
Conjecture: (n+1)*a(n) + 3*(-n+1)*a(n-1) + 2*(-2*n+1)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
Conjecture is true since the g.f. satisfies (x - 3*x^2 - 4*x^3)*g'(x) + (1 - 6*x^2)*g(x) = 1. - Robert Israel, May 22 2015
a(n) = (-1)^n/A001622 + A000108(n+1)*hypergeom([1, n + 3/2], [n + 3], -4). - Vladimir Reshetnikov, Oct 02 2016
a(n) ~ 2^(2*n + 2) / (5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 03 2016
a(n) = (A000108(n) * (2 + (n + 1)*hypergeom([1,-n], [1/2], 5/4)) - (-1)^n)/2. - Vladimir Reshetnikov, Oct 03 2016

More terms from Christian G. Bower, Apr 15 1998

More terms from Alexander Adamchuk, Jul 03 2006

A117641 Number of 3-Motzkin paths of length n with no level steps at height 0.

Original entry on oeis.org

1, 0, 1, 3, 11, 42, 167, 684, 2867, 12240, 53043, 232731, 1031829, 4615542, 20805081, 94410363, 430945739, 1977366192, 9115261211, 42195093993, 196060049129, 914110333422, 4275222950221, 20051858039718, 94294269673861

Offset: 0

Louis Shapiro, Apr 10 2006

Hankel transform of this sequence forms A000012 = [1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007

			The a(4) = 11 paths are UUDD, UDUD and 9 of the form UXYD where each of X and Y are level steps in any of three colors.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
L. W. Shapiro, C. J. Wang, A bijection between 3-Motzkin paths and Schroder paths with no peak at odd height, JIS 12 (2009) 09.3.2.

Cf. A000957, A001006, A002212, A005043, A097331, A000108.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+3*x-Sqrt(1-6*x+5*x^2))/(2*x*(3+x)) )); // G. C. Greubel, Apr 04 2019

CoefficientList[ Series[(1 + 3x - Sqrt[1 - 6x + 5x^2])/(2x^2 + 6x), {x, 0, 25}], x] (* Robert G. Wilson v *)

a(n):=sum(3^(n-2*j)*binomial(n+1,j)*binomial(n-j-1,n-2*j),j,0,floor(n/2))/(n+1); /*  Vladimir Kruchinin, Apr 04 2019 */

my(x='x+O('x^30)); Vec( (1+3*x-sqrt(1-6*x+5*x^2))/(2*x*(3+x)) ) \\ G. C. Greubel, Apr 04 2019

((1+3*x-sqrt(1-6*x+5*x^2))/(2*x*(3+x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 04 2019

G.f.: (1 +3*x -sqrt(1 -6*x +5*x^2))/(2*x*(3+x)).
G.f. as continued fraction is 1/(1-0*x-x^2/(1-3*x-x^2/(1-3*x-x^2/(1-3*x-x^2/(.....))))). - Paul Barry, Dec 02 2008
a(n) = A126970(n,0). - Philippe Deléham, Nov 24 2009
a(n) = Sum_{k=0..n} A091965(n,k)*(-3)^k. - Philippe Deléham, Nov 28 2009
a(n) = Sum_{k=1..n} Sum_{j=0..floor((n-2*k)/2)} 3^(n-2*k-2*j)*(k/(k+2*j))*binomial(k+2*j,j)*binomial(n-k-1,n-2*k-2*j). - José Luis Ramírez Ramírez, Mar 22 2012
D-finite with recurrence: 3*(n+1)*a(n) +(-17*n+10)*a(n-1) +9*(n-3)*a(n-2) +5*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 02 2012
a(n) ~ 5^(n+3/2) / (32 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014
a(n) = 1/(n+1)*Sum_{j=0..floor(n/2)} 3^(n-2*j)*C(n+1,j)*C(n-j-1,n-2*j). - Vladimir Kruchinin, Apr 04 2019

A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

Original entry on oeis.org

1, 1, 7, 64, 678, 7836, 95838, 1219527, 15979551, 214151601, 2921712145, 40444378948, 566634504256, 8019501351103, 114484746457075, 1646614155398872, 23837794992712680, 347081039681365623, 5079306905986689309, 74670702678690897079, 1102218694940440851877

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..836

Cf. A002212, A307678, A349331, A349332, A349334, A349335.

Cf. A002295, A346648 (partial sums), A349362.

a:= n-> coeff(series(RootOf(1+x*A^6/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^6/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^6, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(6*k,k) / (5*k+1).

a(n) ~ 49781^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 3, 0, 5, 0, 1, 0, 0, 0, 1, 0, 6, 0, 6, 0, 1, 0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1, 0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1, 0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1, 0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1, 0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1

Offset: 0

Joerg Arndt, Mar 29 2014

Triangle A129182 transposed.
Column sums give the Catalan numbers (A000108).
Row sums give A143951.
Sums along falling diagonals give A005169.
T(4n,2n) = A240008(n). - Alois P. Heinz, Mar 30 2014

			Triangle begins:
00:  1;
01:  0, 1;
02:  0, 0, 1;
03:  0, 0, 0, 1;
04:  0, 0, 1, 0, 1;
05:  0, 0, 0, 2, 0, 1;
06:  0, 0, 0, 0, 3, 0, 1;
07:  0, 0, 0, 1, 0, 4, 0, 1;
08:  0, 0, 0, 0, 3, 0, 5, 0, 1;
09:  0, 0, 0, 1, 0, 6, 0, 6, 0, 1;
10:  0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1;
11:  0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1;
12:  0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1;
13:  0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1;
14:  0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1;
15:  0, 0, 0, 0, 0, 5, 0, 35, 0, 63, 0, 45, 0, 12, 0, 1;
16:  0, 0, 0, 0, 1, 0, 16, 0, 65, 0, 92, 0, 55, 0, 13, 0, 1;
17:  0, 0, 0, 0, 0, 5, 0, 40, 0, 112, 0, 129, 0, 66, 0, 14, 0, 1;
18:  0, 0, 0, 0, 0, 0, 16, 0, 86, 0, 182, 0, 175, 0, 78, 0, 15, 0, 1;
19:  0, 0, 0, 0, 0, 3, 0, 43, 0, 167, 0, 282, 0, 231, 0, 91, 0, 16, 0, 1;
20:  0, 0, 0, 0, 0, 0, 14, 0, 102, 0, 301, 0, 420, 0, 298, 0, 105, 0, 17, 0, 1;
...
Column k=4 corresponds to the following 14 paths (dots denote zeros):
#:         path              area   steps (Dyck word)
01:  [ . 1 . 1 . 1 . 1 . ]     4     + - + - + - + -
02:  [ . 1 . 1 . 1 2 1 . ]     6     + - + - + + - -
03:  [ . 1 . 1 2 1 . 1 . ]     6     + - + + - - + -
04:  [ . 1 . 1 2 1 2 1 . ]     8     + - + + - + - -
05:  [ . 1 . 1 2 3 2 1 . ]    10     + - + + + - - -
06:  [ . 1 2 1 . 1 . 1 . ]     6     + + - - + - + -
07:  [ . 1 2 1 . 1 2 1 . ]     8     + + - - + + - -
08:  [ . 1 2 1 2 1 . 1 . ]     8     + + - + - - + -
09:  [ . 1 2 1 2 1 2 1 . ]    10     + + - + - + - -
10:  [ . 1 2 1 2 3 2 1 . ]    12     + + - + + - - -
11:  [ . 1 2 3 2 1 . 1 . ]    10     + + + - - - + -
12:  [ . 1 2 3 2 1 2 1 . ]    12     + + + - - + - -
13:  [ . 1 2 3 2 3 2 1 . ]    14     + + + - + - - -
14:  [ . 1 2 3 4 3 2 1 . ]    16     + + + + - - - -
There are no paths with weight < 4, one with weight 4, none with weight 5, 3 with weight 6, etc., therefore column k=4 is
[0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, ...].
Row n=8 is [0, 0, 0, 0, 3, 0, 5, 0, 1], the corresponding paths of weight=8 are:
Semilength 4:
  [ . 1 . 1 2 1 2 1 . ]
  [ . 1 2 1 . 1 2 1 . ]
  [ . 1 2 1 2 1 . 1 . ]
Semilength 6:
  [ . 1 . 1 . 1 . 1 . 1 2 1 . ]
  [ . 1 . 1 . 1 . 1 2 1 . 1 . ]
  [ . 1 . 1 . 1 2 1 . 1 . 1 . ]
  [ . 1 . 1 2 1 . 1 . 1 . 1 . ]
  [ . 1 2 1 . 1 . 1 . 1 . 1 . ]
Semilength 8:
  [ . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . ]

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Sequences obtained by particular choices for x and y in the g.f. F(x,y) are: A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(sqrt(x), sqrt(x))),  A227310 (1+x*F(x, x^2), also 2-1/F(x, 1)), A239928 (F(x^2, x)), A052709 (x*F(1,x+x^2)), A125305 (F(1, x+x^3)), A002212 (F(1, x/(1-x))).
Cf. A047998, A138158, A227543.
Cf. A129181.

b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0, 0, `if`(x=0, `if`(k=0, 1, 0),
       b(x-1, y-1, k-y+1/2)+ b(x-1, y+1, k-y-1/2)))
    end:
T:= (n, k)-> b(2*k, 0, n):
seq(seq(T(n, k), k=0..n), n=0..20);  # Alois P. Heinz, Mar 29 2014

b[x_, y_, k_] := b[x, y, k] = If[y<0 || y>x || k<0, 0, If[x == 0, If[k == 0, 1, 0], b[x-1, y-1, k-y+1/2] + b[x-1, y+1, k-y-1/2]]]; T[n_, k_] := b[2*k, 0, n]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }
print_triangle(V)= { my( N=#V ); for(n=1, N, print( rvec( V[n]) ) ); }
N=20; x='x+O('x^N);
F(x,y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );
v= Vec( F(x,y) );
print_triangle(v)

G.f.: F(x,y) satisfies F(x,y) = 1 / (1 - x*y * F(x, x^2*y) ).

G.f.: 1/(1 - y*x/(1 - y*x^3/(1 - y*x^5/(1 - y*x^7/(1 - y*x^9/( ... )))))).

A026118 Number of polyhexes of class PF2 (with two catafusenes annealated to pyrene).

Original entry on oeis.org

5, 20, 100, 431, 1937, 8548, 38199, 171001, 770934, 3492251, 15905897, 72785480, 334571647, 1544203452, 7154247842, 33260560977, 155126129968, 725639264293, 3403612632885, 16004969728270, 75437244856898, 356337397010035, 1686618801843050

Offset: 6

N. J. A. Sloane

nonn

See reference for precise definition.
From Petros Hadjicostas, Jan 13 2019: (Start)
This sequence is defined by eq. (34), p. 536, in Cyvin et al. (1992). It is denoted by 2^Q_{4+n} (for n >= 2). Thus, a(n+4) = 2^Q_{4+n} for n >= 2 (and that is why the offset here is 6).
For n >= 2, we have a(n+4) = (3/4)*(1 + (-1)^n)*N(floor(n/2)) + (1/4)*(L(n) + 13*Sum_{1 <= i <= n-1} N(i)*N(n-i)), where N(n) = A002212(n) and L(n) = A039658(n).
The sequence (N(n): n >= 1) = (A002212(n): n >= 1) is given by eq. (1), p. 533, in Cyvin et al. (1992), while its g.f. is given by eqs. (2)-(4), p. 1174, in Cyvin et al. (1994). (The g.f. of N(n) = A002212(n) appears also in Harary and Read (1970) as eq. (9) on p. 4.)
The sequence (L(n): n >= 1) = (A039658(n): n >= 1) is given by eq. (22), p. 535, in Cyvin et al (1992), while its g.f. is given by eq. (9), p. 1175, in Cyvin et al. (1994).
The g.f. of the current sequence (a(m): m >= 6) (see below) is given in eq. (A2), p. 1180, in Cyvin et al. (1994), but it can be derived by the above formulae using standard techniques for the calculation of g.f.'s.
For the number of polyhexes of class PF2, we have 1^Q_h = A026106(h) (h >= 5, one catafusene annealated to pyrene), 3^Q_h = A026298(h) (h >= 7, three catafusenes annealated to pyrene), and 4^Q_h = A030519(h) (h >= 8, four catafusenes annealated to pyrene).
(Apparently, the word "annealated" in Cyvin et al. (1992) is spelled "annelated" in Cyvin et al. (1994).)
(End)

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
Eric Weisstein's World of Mathematics, Fusenes.
Eric Weisstein's World of Mathematics, Polyhex.

Cf. A002212, A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534, A039658.

From Petros Hadjicostas, Jan 14 2019: (Start)
a(n+4) = (3/4)*(1 + (-1)^n)*N(floor(n/2)) + (1/4)*(L(n) + 13*Sum_{1 <= i <= n-1} N(i)*N(n-i)) for n >= 2, where N(n) = A002212(n) and L(n) = A039658(n).
G.f.: (x^2/4)*(1-x)^(-1)*(10 - 48*x + 74*x^2 - 38*x^3) - (x^2/8)*[13*(1 - 3*x)*(1 - x)^(1/2)*(1 - 5*x)^(1/2) + (1 - x)^(-1)*(7 - 5*x)*(1 - x^2)^(1/2)*(1 - 5*x^2)^(1/2)] (see eq. (A2), p. 1180, in Cyvin et al. (1994)).
(End)

Name edited by Petros Hadjicostas, Jan 13 2019

Terms a(17)-a(28) computed by Petros Hadjicostas, Jan 13 2019 using a g.f. in Cyvin et al. (1994)

A026759 a(n) = T(2n, n), T given by A026758.

Original entry on oeis.org

1, 2, 7, 27, 109, 453, 1922, 8284, 36155, 159435, 709246, 3178992, 14343567, 65099245, 297015765, 1361584755, 6268757195, 28975155915, 134410918700, 625578384150, 2920488902795, 13672762887465, 64179220019365, 301987822527627

Offset: 0

Clark Kimberling

nonn

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

Cf. A026758, A026760, A026761, A026762, A026763, A026764, A026765, A026766, A026767, A026768.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( ((1-x)*Sqrt(1 - 4*x) - Sqrt(1 - 6*x + 5*x^2))/(2*x^2) )); // G. C. Greubel, Oct 31 2019

seq(coeff(series(((1-x)*sqrt(1-4*x) - sqrt(1 -6*x +5*x^2))/(2*x^2), x, n+2), x, n), n = 0..30); # G. C. Greubel, Oct 31 2019

CoefficientList[Normal[Series[((1-x)Sqrt[1-4x] -Sqrt[1-6x+5x^2])/(2x^2), {x, 0, 30}]], x] (* David Callan, Feb 01 2014 *)

my(x='x+O('x^30)); Vec(((1-x)*sqrt(1 - 4*x) - sqrt(1 - 6*x + 5*x^2))/(2*x^2)) \\ G. C. Greubel, Oct 31 2019

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(((1-x)*sqrt(1-4*x) - sqrt(1-6*x+5*x^2))/(2*x^2)).list()
A077952_list(30) # G. C. Greubel, Oct 31 2019

a(n) = A002212(n+1) - A000245(n). - David Callan, Feb 01 2014

G.f.: ((1-x)*sqrt(1 - 4*x) - sqrt(1 - 6*x + 5*x^2))/(2*x^2). - G. C. Greubel, Oct 31 2019

A030532 Number of polyhexes of class PF2 with symmetry point group C_s.

Original entry on oeis.org

0, 1, 6, 35, 168, 807, 3738, 17326, 79909, 369330, 1709087, 7929590, 36880231, 171981241, 804008476, 3767969067, 17699758030, 83328230588, 393123455667, 1858351021018, 8801159427825, 41756067216508, 198437454009869, 944521139813575, 4502419756667924

Offset: 4

N. J. A. Sloane

nonn

See reference for precise definition.

Cyvin has incorrect a(13)=369366 and a(14)=1709123 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II.

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
Sean A. Irvine, Java program (github)

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

L(n) = my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*x^2*(1-x)), n); \\ A039658
Lp(n) = my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-6*x^2+7*x^4-(1-3*x^2)*sqrt(1-6*x^2+5*x^4))/(2*x^4*(1-x)), n); \\ A039660
M(n)= my(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)); \\ A055879
N(n) = polcoeff( (1 - x - sqrt(1 - 6*x + 5*x^2 + x^2 * O(x^n))) / 2, n+1); \\ A002212
Mp(n) = N(n) - sum(j=0, n-1, N(j)); \\ A039919
b(n) = N(n+3) - 6*N(n+2) - Mp(floor((n+1)/2)) + (41*N(n+1)-21*N(n)-L(n))/4 - (M(n+3)-M(n+2)+M(n)-if (!(n%2),M(n/2))+Lp(n))/2;
a(n) = if (n<=4, 0, b(n-4)); \\ Michel Marcus, Apr 05 2020

a(n+4) = N(n+3) - 6*N(n+2) - M'(floor((n+1)/2)) + (41*N(n+1)-21*N(n)-L(n))/4 - (M(n+3)-M(n+2)+M(n)-e(n)*M(n/2)+L'(n))/2 where N(n)=A002212(n), M(n)=A055879(n), M'(n)=A039919(n), L(n)=A039658(n), L'(n)=A039660(n), e(n)=1 if n is even and 0 if n is odd. - Sean A. Irvine, Apr 03 2020

a(13) and a(14) corrected, title improved, and more terms from Sean A. Irvine, Apr 03 2020

A045868 Expansion of g.f.: ((1 - x - sqrt(1-6x+5x^2))/(2*x))^2.

Original entry on oeis.org

1, 2, 7, 26, 101, 406, 1676, 7066, 30302, 131782, 579867, 2576982, 11550237, 52152330, 237005385, 1083211410, 4975796735, 22960105510, 106377393365, 494674698190, 2308015808015, 10801388134690, 50691017885290, 238503869991926, 1124828963516896, 5316520644648026, 25179670936870021

Offset: 0

N. J. A. Sloane

Convolution of A002212 with itself.
Number of skew Dyck paths of semilength n+1 starting with UU. 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)=7 because we have UUDDUD, UUDUDD, UUDUDL, UUUDDD, UUUDDL, UUUDLD and UUUDLL. - Emeric Deutsch, May 11 2007
a(n) is also the number of path-pairs (u,v) having the following six properties: 1) the lengths of u and v sum up to 2n, 2) u and v both start at (0,0), 3) (0,0) is the only vertex that u and v have in common, 4) the steps that u can make are (1,0), (0,1) and (0,-1), 5) the steps that v can make are (1,0), (-1,0) and (0,1), 6) if A and B are the termini of u and v, respectively, then B=A+(1,-1). - Svjetlan Feretic, Jun 09 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
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. 99.

Cf. A055450.

Essentially the first differences of A002212 and A025238.

[n le 2 select n else (2*(3*n-2)*Self(n-1) - 5*(n-3)*Self(n-2))/(n+1): n in [1..30]]; // G. C. Greubel, Jan 12 2024

a := n->(2/n)*sum(binomial(n,j)*binomial(2*j+1,j-1),j=1..n): 1,seq(a(n),n=1..22);

a[n_] := 2*Hypergeometric2F1[ 5/2, 1-n, 4, -4]; a[0] = 1; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 30 2012, after Maple *)

a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))^2/4,n+2)

my(x='x+O('x^66)); Vec(((1-x-sqrt(1-6*x+5*x^2))/(2*x))^2) \\ Joerg Arndt, May 04 2013

def A045868(n): return 1 if n==0 else (2/n)*sum( binomial(n,j)*binomial(2*j+1,j-1) for j in range(1,n+1))
[A045868(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

a(n) = (2/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+1, j-1) for n >= 1.
a(n) = A055450(n, n-1).
D-finite with recurrence: (n+2)*a(n) = (6*n+2)*a(n-1) - (5*n-10)*a(n-2). - Vladeta Jovovic, Jul 16 2004
a(n) = 2*Hypergeometric2F1(5/2, 1-n, 4, -4). - Jean-François Alcover, Apr 30 2012
a(n) ~ 2*5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
G.f.: 1 - 1/x + Q(0)*(1-x)/x, where Q(k) = 1 + (4*k+1)*x/((1-x)*(k+1) - x*(1-x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
G.f.: 1/x - 1 - 2*(1-x)/x/( G(0) + 1), where G(k) = 1 + 2*x*(4*k+1)/( (2*k+1)*(1-x) - x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013

More terms from Emeric Deutsch, May 11 2007

cp's OEIS Frontend

A130777 Coefficients of first difference of Chebyshev S polynomials.

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A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells.

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A032357 Convolution of Catalan numbers and powers of -1.

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A117641 Number of 3-Motzkin paths of length n with no level steps at height 0.

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A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

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A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

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A026118 Number of polyhexes of class PF2 (with two catafusenes annealated to pyrene).

A045868 Expansion of g.f.: ((1 - x - sqrt(1-6x+5x^2))/(2*x))^2.