A002212 -id:A002212 - cp's OEIS frontend

A104259 Triangle T read by rows: matrix product of Pascal and Catalan triangle.

1, 2, 1, 5, 4, 1, 15, 14, 6, 1, 51, 50, 27, 8, 1, 188, 187, 113, 44, 10, 1, 731, 730, 468, 212, 65, 12, 1, 2950, 2949, 1956, 970, 355, 90, 14, 1, 12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1, 51822, 51821, 35643, 19474, 8612, 3021, 805, 152, 18, 1

Offset: 0

Ralf Stephan, Mar 17 2005

Also, Riordan array (G,G), G(t)=(1 - ((1-5*t)/(1-t))^(1/2))/(2*t).
From Emanuele Munarini, May 18 2011: (Start)
Row sums = A002212.
Diagonal sums = A190737.
Central coefficients = A190738. (End)

			Triangle begins:
1
2, 1
5, 4, 1
15, 14, 6, 1
51, 50, 27, 8, 1
188, 187, 113, 44, 10, 1
731, 730, 468, 212, 65, 12, 1
2950, 2949, 1956, 970, 355, 90, 14, 1
12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1
Production matrix begins
2, 1
1, 2, 1
1, 1, 2, 1
1, 1, 1, 2, 1
1, 1, 1, 1, 2, 1
1, 1, 1, 1, 1, 2, 1
1, 1, 1, 1, 1, 1, 2, 1
... - _Philippe Deléham_, Mar 01 2013

Robert Israel, Table of n, a(n) for n = 0..5049
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [alternative link]
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.

T = A007318 * A033184.

Left-hand columns include A007317, A007317 - 1. Row sums are in A002212.

T := (n,k) -> binomial(n,k)*hypergeom([k/2+1/2,k/2+1,k-n],[k+1,k+2],-4); seq(print(seq(round(evalf(T(n,k),99)),k=0..n)),n=0..8); # Peter Luschny, Sep 23 2014
# Alternative:
N:= 12:  # to get the first N rows
P:= Matrix(N,N,(i,j) -> binomial(i-1,j-1), shape=triangular[lower]):
C:= Matrix(N,N,(i,j) -> binomial(2*i-j-1,i-j)*j/i, shape=triangular[lower]):
T:= P . C:
for i from 1 to N do
seq(T[i,j],j=1..i)
od;   # Robert Israel, Sep 23 2014

Flatten[Table[Sum[Binomial[n,i]Binomial[2i-k,i-k](k+1)/(i+1),{i,k,n}],{n,0,100},{k,0,n}]] (* Emanuele Munarini, May 18 2011 *)

create_list(sum(binomial(n,i)*binomial(2*i-k,i-k)*(k+1)/(i+1),i,k,n),n,0,12,k,0,n); /* Emanuele Munarini, May 18 2011 */

T(n,k) = sum(binomial(n,i)*binomial(2*i-k,i-k)*(k+1)/(i+1),i=k..n).
T(n+1,k+2) = T(n+1,k+1) + T(n,k+2) - T(n,k+1) - T(n,k). - Emanuele Munarini, May 18 2011
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + Sum_{i, i>=0} T(n-1,k+1+i). - Philippe Deléham, Feb 23 2012
T(n,k) = C(n,k)*hypergeom([k/2+1/2,k/2+1,k-n],[k+1,k+2],-4). - Peter Luschny, Sep 23 2014

A026106 Number of polyhexes of class PF2 (with one catafusene annealated to pyrene).

Original entry on oeis.org

2, 5, 16, 55, 208, 817, 3336, 13935, 59406, 257079, 1126948, 4992421, 22318048, 100546543, 456055730, 2080872845, 9544572590, 43984730855, 203550840696, 945562887981, 4407586685688, 20609668887723, 96646196091276, 454402001079165

Offset: 5

N. J. A. Sloane

nonn

See reference for precise definition.
From Petros Hadjicostas, Jan 12 2019: (Start)
In Cyvin et al. (1992), sequence (N(m): m >= 1) = (A002212(m): m >= 1) is defined by eq. (1), p. 533. (We may let N(0) := A002212(0) = 1.)
Sequence (M(m): m >= 1) is defined by eq. (13), p. 534. We have M(2*m) = M(2*m-1) =  A007317(m) for m >= 1.
Sequences (N(m): m >= 1) and (M(m): m >= 1) appear in Table 1, p. 533.
The current sequence is denoted by 1^Q_(4+n) (with n = 1,2,3,...). Thus, a(n+4) = 1^Q_(4+n) for n >= 1; i.e., a(m) = 1^Q_{m} for m >= 5. We have 1^Q_(4+n) = (1/2)*(3*N(n) + M(n)) for n >= 1. See eq. (33), p. 536.
Sequence (1^Q_(4+n): n >= 1) appears in Table II, p. 537.
We may use the many formulae in the documentations of sequences A002212 and A007317 in order to create complicated formulae and recurrence relations for (a(n): n >= 5). We omit the details.
The first g.f. below is a combination of the g.f. for sequence A002212 by John W. Layman in 2001 and the g.f. for sequence A007317 by Ira M. Gessel and Jang Soo Kim in 2010.
The second g.f. appears in eq. (A1), p. 1180, in Cyvin et al. (1994). It is algebraically equivalent to the first g.f.
(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.
Eric Weisstein's World of Mathematics, Fusenes.
Eric Weisstein's World of Mathematics, Polyhex.

Cf. A002212, A007317, A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534, A039658.

bb := proc(x) (1/4)*x^3*(4-8*x-3*sqrt((1-x)*(1-5*x))-(x+1)*sqrt((1-5*x^2)/(1-x^2))) end proc;
taylor(bb(x), x = 0, 50); # Petros Hadjicostas, Jan 12 2019

(1/4) x^3 (4 - 8x - 3Sqrt[(1-x)(1-5x)] - (x+1) Sqrt[(1-5x^2)/(1-x^2)]) + O[x]^29 // CoefficientList[#, x]& // Drop[#, 5]& (* Jean-François Alcover, Apr 24 2020, from Maple *)

From Petros Hadjicostas, Jan 12 2019: (Start)
For n >= 1, a(n+4) = (1/2)*(3*A002212(n) + A007317(floor((n+1)/2))).
G.f.: (x^3/4)*(4 - 8*x - 3*sqrt(1 - 6*x + 5*x^2) - (x + 1)*sqrt((1 - 5*x^2)/(1 - x^2))).
G.f.: x^3*(1 - 2*x) - (x^3/4)*(3*(1 - x)^(1/2)*(1 - 5*x)^(1/2) + (1 - x)^(-1)*(1 - x^2)^(1/2)*(1 - 5*x^2)^(1/2)) (see eq. (A1), p. 1180, in Cyvin et al. (1994)).
(End)

Name edited by Petros Hadjicostas, Jan 12 2019

Terms a(17)-a(28) computed by Petros Hadjicostas, Jan 12 2019

A026298 Number of polyhexes of class PF2.

Original entry on oeis.org

4, 28, 176, 950, 4908, 24402, 119240, 575348, 2757460, 13157752, 62638788, 297832008, 1415550920, 6728600060, 31998023632, 152271569872, 725231959452, 3457304575812, 16497751608120, 78804354881238, 376806016649964, 1803539487096138, 8641075826669256, 41441524062045660

Offset: 7

N. J. A. Sloane

nonn

See reference for precise definition.

Cyvin et al. has incorrect a(13) = 119204 and a(14) = 575312 due to using incorrect value for A039919(5); cf. A039659. - Sean A. Irvine, Sep 24 2019

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.

Cf. A026106, A026118, A030519, A030520, A030525, A030529, A030532, A030534, A002212, A039919.

a(n + 4) = 3 * (N(n+2) - 6*N(n+1) + 8*N(n)) + A039919(floor((n+1)/2)) where N(n) = A002212(n) [from Cyvin]. - Sean A. Irvine, Sep 24 2019

Corrected and extended by Sean A. Irvine, Sep 24 2019

A030519 Number of polyhexes of class PF2 with four catafusenes annealated to pyrene.

Original entry on oeis.org

2, 13, 101, 619, 3641, 20028, 106812, 554352, 2828660, 14244878, 71077246, 352184306, 1736118578, 8525182798, 41741378126, 203929434766, 994680883360, 4845761306611, 23586192274443, 114731539477465, 557859497501007, 2711772157178038, 13180227306740726

Offset: 8

N. J. A. Sloane

nonn

See reference for precise definition.

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.

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
b(n) = N(n+3) - 9*N(n+2) + 25*N(n+1) - 21*N(n) + (M(n+3) - M(n+2) - 3*M(n+1) + 3*M(n) + Lp(n))/2;
a(n) = b(n-4); \\ Michel Marcus, Apr 03 2020

a(n+4) = N(n+3) - 9*N(n+2) + 25*N(n+1) - 21*N(n) + (M(n+3) - M(n+2) - 3*M(n+1) + 3*M(n) + L'(n))/2 where N(n)=A002212(n), M(n)=A055879(n), and L'(n)=A039660(n) for n >= 4. - Sean A. Irvine, Apr 02 2020

More terms and title improved by Sean A. Irvine, Apr 02 2020

A182401 Number of paths from (0,0) to (n,0), never going below the x-axis, using steps U=(1,1), H=(1,0) and D=(1,-1), where the H steps come in five colors.

Original entry on oeis.org

1, 5, 26, 140, 777, 4425, 25755, 152675, 919139, 5606255, 34578292, 215322310, 1351978807, 8550394455, 54419811354, 348309105300, 2240486766555, 14476490777175, 93914850905862, 611489638708140, 3994697746533171, 26175407271617955, 171991872078871311

Offset: 0

Emanuele Munarini, Apr 27 2012

Number of 3-colored Schroeder paths from (0,0) to (2n+2,0) with no level steps H=(2,0) at even level. H-steps at odd levels are colored with one of the three colors. Example: a(2)=5 because we have UUDD, UHD (3 choices) and UDUD. - José Luis Ramírez Ramírez, Apr 27 2015

			seq(3^n * simplify(hypergeom([3/2, -n], [3], -4/3)), n = 0..20); # _Peter Bala_, Feb 04 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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. A001006, A002212, A005572, A025230, A126120, A257290.

CoefficientList[Series[(1-5*x-Sqrt[1-10*x+21*x^2])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := 5^n*Hypergeometric2F1[(1-n)/2, -n/2, 2, 4/25]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 22 2013, after 2nd formula *)

a(n):=coeff(expand((1+5*x+x^2)^(n+1)),x^n)/(n+1);
makelist(a(n),n,0,30);

x='x+O('x^66); Vec((1-5*x-sqrt(1-10*x+21*x^2))/(2*x^2)) \\ Joerg Arndt, Jun 02 2013

a(n) = [x^n] (1+5*x+x^2)^(n+1)/(n+1).
a(n) = Sum_{k=0..floor(n/2)} (binomial(n,2*k)*binomial(2*k,k)/(k+1))*5^(n-2*k).
G.f.: (1-5*x-sqrt(1-10*x+21*x^2))/(2*x^2).
Conjecture: (n+2)*a(n) +5*(-2*n-1)*a(n-1) +21*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 24 2012
a(n) ~ 7^(n+3/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
a(n) = A125906(n,0). - Philippe Deléham, Mar 04 2013
G.f.: 1/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017
From Seiichi Manyama, Jan 15 2024: (Start)
G.f.: (1/x) * Series_Reversion( x / (1+5*x+x^2) ).
a(n) = (1/(n+1)) * Sum_{k=0..n} 3^(n-k) * binomial(n+1,n-k) * binomial(2*k+2,k). (End)
From Peter Bala, Feb 03 2024: (Start)
G.f: 1/(1 - 3*x)*c(x/(1 - 3*x))^2 = 1/(1 - 7*x)*c(-x/(1 - 7*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
a(n) = Sum_{k = 0..n} 3^(n-k)*binomial(n, k)*Catalan(k+1).
a(n) = 3^n * hypergeom([3/2, -n], [3], -4/3).
a(n) = 7^n * Sum_{k = 0..n} (-7)^(-k)*binomial(n, k)*Catalan(k+1).
a(n) = 7^n * hypergeom([3/2, -n], [3], 4/7). (End)

A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

Original entry on oeis.org

1, 1, 9, 109, 1541, 23823, 390135, 6651051, 116798643, 2098313686, 38382509118, 712447023590, 13385500614902, 254065657922154, 4864482597112186, 93840443376075810, 1822169236520766546, 35586928273002974487, 698572561837366684479, 13775697096997873764647

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

In general, for m > 1, Sum_{k=0..n} binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (1 + m^m/(m-1)^(m-1))^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - Vaclav Kotesovec, Nov 15 2021

Seiichi Manyama, Table of n, a(n) for n = 0..756

Cf. A002212, A307678, A349331, A349332, A349333, A349334.

Cf. A007556, A346650 (partial sums), A349364.

a:= n-> coeff(series(RootOf(1+x*A^8/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 15 2021

nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^8/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^8, 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(8*k,k) / (7*k+1).

a(n) ~ 17600759^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A055450 Path-counting array T; each step of a path is (1 right) or (1 up) to a point below line y=x, else (1 right and 1 up) or (1 up) to a point on the line y=x, else (1 left) or (1 up) to a point above line y=x. T(i,j)=number of paths to point (i-j,j), for 1<=j<=i, i >= 1.

Original entry on oeis.org

1, 1, 3, 1, 2, 10, 1, 3, 7, 36, 1, 4, 5, 26, 137, 1, 5, 9, 19, 101, 543, 1, 6, 14, 14, 75, 406, 2219, 1, 7, 20, 28, 56, 305, 1676, 9285, 1, 8, 27, 48, 42, 230, 1270, 7066, 39587, 1, 9, 35, 75, 90, 174, 965, 5390, 30302, 171369, 1, 10, 44, 110, 165, 132, 735, 4120, 23236, 131782, 751236

Offset: 0

Clark Kimberling, May 18 2000

			Triangle begins as:
  1;
  1, 3;
  1, 2, 10;
  1, 3,  7, 36;
  1, 4,  5, 26, 137;
  1, 5,  9, 19, 101, 543;
  1, 6, 14, 14,  75, 406, 2219;
  1, 7, 20, 28,  56, 305, 1676, 9285;
  1, 8, 27, 48,  42, 230, 1270, 7066, 39587;
  ...
T(4,4) defined as T(5,4)+T(3,3) when k=4, T(5,4) already defined when k=3.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000108, A002212, A030237, A045868,

Cf. A055451, A055452, A055453, A055454, A055455.

B:=Binomial; G:=Gamma; F:=Factorial;
p:= func< n,k,j | B(n-2*k+j-1, j)*G(n-k+j+3/2)/(F(j)*G(n-k+3/2)*B(n-k+j+2, j)) >;
A030237:= func< n,k | (n-k+1)*Binomial(n+k, k)/(n+1) >;
function T(n,k) // T = A055450
  if k lt n/2 then return A030237(n-k+1, k);
  else return Round(Catalan(n-k+1)*(&+[p(n,k,j)*(-4)^j: j in [0..n]]));
  end if;
end function;
[T(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 29 2024

T[n_, 0]:= 1; T[n_, k_]:= T[n, k]= If[1<=kG. C. Greubel, Jan 29 2024 *)
T[n_, k_]:= If[kG. C. Greubel, Jan 29 2024 *)

def A030237(n,k): return (n-k+1)*binomial(n+k, k)/(n+1)
def T(n,k): # T = A055450
    if kA030237(n-k+1,k)
    else: return round(catalan_number(n-k+1)*hypergeometric([n-2*k, (3+2*(n-k))/2], [3+n-k], -4))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 29 2024

Initial values: T(i, 0)=1 for i >= 0. Recurrence: if 1 <= j < i/2, then T(i, j) = T(i-1, j-1) + T(i-1, j), if j = i/2 then T(2j, j) = T(2j-2, j-1) + T(2j-1, j-1), otherwise T(2j-k, j) = T(2j-k+1, j) + T(2j-k-1, j-1) for j=k, k+1, k+2, ..., for k=1, 2, 3, ...
T(2n, n) = A000108(n) for n >= 0 (Catalan numbers).
T(n, n) = A002212(n+1).
T(n, n-1) = A045868(n).
T(n, k) = A030237(n-k+1, k) for n >= 1, 0 <= k < n/2.
From G. C. Greubel, Jan 29 2024: (Start)
T(n, k) = (n-2*k+2)*binomial(n+1, k)/(n-k+2) for 0 <= k < n/2, otherwise Catalan(n-k +1)*Hypergeometric2F1([n-2*k, (3+2*(n-k))/2], [3+n-k], -4).
Sum_{k=0..n} T(n, k) = A055451(n). (End)

A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

Original entry on oeis.org

1, 1, 8, 85, 1051, 14197, 203064, 3022909, 46347534, 726894786, 11606936525, 188060979332, 3084087347910, 51094209834068, 853859480938095, 14376597494941454, 243649099741045190, 4153091242153905838, 71152973167920086796, 1224593757045581062444

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..791

Cf. A002212, A307678, A349331, A349332, A349333, A349335.

Cf. A002296, A346649 (partial sums), A349363.

a:= n-> coeff(series(RootOf(1+x*A^7/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^7/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^7, 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(7*k,k) / (6*k+1).

a(n) ~ 870199^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A360100 a(n) = Sum_{k=0..n} binomial(n+2k-1,n-k) Catalan(k).

Original entry on oeis.org

1, 1, 5, 23, 111, 562, 2952, 15948, 88076, 495077, 2823293, 16295020, 95007654, 558765743, 3310999269, 19748462718, 118471172054, 714355994997, 4327148812557, 26319195869861, 160677354596769, 984236344800234, 6047526697800992, 37262944840704171

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Partial sums are A360102.
Cf. A162481, A258973.
Cf. A002212, A006319, A162475, A360101.
Cf. A000108.

A360100 := proc(n)
    add(binomial(n+2*k-1,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360100(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

m = 24;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^3 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)

a(n) = sum(k=0, n, binomial(n+2*k-1, n-k)*binomial(2*k, k)/(k+1));

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^3)))

G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^3.
G.f.: c(x/(1-x)^3), where c(x) is the g.f. of A000108.
a(n) ~ sqrt(-2 + (35 - 3*sqrt(129))^(1/3) + (35 + 3*sqrt(129))^(1/3)) * (((7 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / (sqrt(2*Pi) * n^(3/2))). - Vaclav Kotesovec, Feb 18 2023
D-finite with recurrence (n+1)*a(n) +(-8*n+5)*a(n-1) +(10*n-27)*a(n-2) +(-4*n+17)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A039919 Related to enumeration of edge-rooted catafusenes.

Original entry on oeis.org

0, 1, 5, 21, 86, 355, 1488, 6335, 27352, 119547, 528045, 2353791, 10575810, 47849685, 217824285, 996999525, 4585548680, 21182609875, 98236853415, 457211008415, 2134851575050, 9997848660345, 46949087361550, 221022160284101, 1042916456739696, 4931673470809525, 23367060132453323

Offset: 1

N. J. A. Sloane

Binomial transform of the first differences of the Catalan numbers (see A000245). - Paul Barry, Feb 16 2006
Starting (1, 5, 21, ...) = A002212, (1, 3, 10, 36, 137, ...) convolved with A007317, (1, 2, 5, 15, 51, ...). - Gary W. Adamson, May 19 2009
From Petros Hadjicostas, Jan 15 2019: (Start)
In Cyvin et al. (1992), sequence (N(m): m >= 1) = (A002212(m): m >= 1) is defined by eq. (1), p. 533. (We may let N(0) := A002212(0) = 1.)
In the same reference, sequence (M(m): m >= 1) is defined by eq. (13), p. 534. We have M(2*m) = M(2*m-1) =  A007317(m) for m >= 1.
In the same reference, the sequence (M'(m): m >= 3) is defined by eq. (26), p. 535; see also Cyvin et al. (1994, Monatshefte fur Chemie), eq. 5, p. 1329. We have M'(m) = Sum_{1 <= i <= floor((m-1)/2)} N(i)*M(m-2*i) for m >= 3.
It turns out that M'(m) = a(floor((m + 1)/2)) for m >= 3, where (a(n): n >= 1) is the current sequence.
If 1 + U(x) = Sum_{n >= 0} N(n)*x^n = Sum_{n >= 0} A002212(n)*x^n, then the g.f. of the sequence (M(m): m >= 1) is V(x) = x*(1-x)^(-1)*(1 + U(x^2)). See eqs. 3 and 4, p. 1329, in Cyvin et al. (1994, Monatshefte fur Chemie).
Eq. 6 in the latter reference (pp. 1329-1330) states that the g.f. of the sequence (M'(m): m >= 3) is U(x^2)*V(x) = U(x^2)*x*(1-x)^(-1)*(1 + U(x^2)).
Since M'(m) = a(floor((m + 1)/2)) for m >= 3, the latter g.f. also equals (1 + x)*A(x^2)/x, where A(x) = Sum_{n >= 1} a(n)*x^n is the g.f. of the current sequence (given below by Emeric Deutsch).
Equating the two forms of the g.f. of the (M'(m): m >= 3), we get that A(x) = x*U(x)*(1 + U(x))/(1-x), where 1 + U(x) is the g.f. of A002212 (with U(0) = 0).
The sequence (M'(m): m >= 3) = (a(floor((m + 1)/2)): m >= 3) is used in the calculation of A026298 (= numbers of polyhexes of the class PF2 with three catafusenes annelated to pyrene).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..200
B. N. Cyvin, E. Brendsdal, J. Brunvoll, and S. J. Cyvin, A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337 (see Eq. 6 for the g.f. of the sequence (M'(n): n >= 3) = (a(floor((m + 1)/2)): m >= 3)).
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.

Cf. A002212, A026298.

Cf. A007317. - Gary W. Adamson, May 19 2009

Table[SeriesCoefficient[8x^2*(1-x)/(1-x+Sqrt[1-6x+5x^2])^3,{x,0,n}],{n,1,23}] (* Vaclav Kotesovec, Oct 08 2012 *)

x='x+O('x^66); concat([0],Vec(8*x^2*(1-x)/(1-x+sqrt(1-6*x+5*x^2))^3)) \\ Joerg Arndt, May 04 2013

G.f.: 8*x^2*(1-x)/(1 - x + sqrt(1 - 6*x + 5*x^2))^3. - Emeric Deutsch, Oct 24 2002
a(n) = A002212(n) - Sum_{j=0..n-1} A002212(j). Example: a(5) = 137 - (1 + 1 + 3 + 10 + 36) = 86. - Emeric Deutsch, Jan 23 2004
a(n+1) = Sum_{k=0..n} C(n,k)*(C(k+1) - C(k)) for n >= 0, where C(k) = A000108(k). - Paul Barry, Feb 16 2006 [edited by Petros Hadjicostas, Jan 18 2019]
Recurrence: (n-2)*(n+1)*a(n) = 2*(n-1)*(3*n-4)*a(n-1) - 5*(n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 3*5^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
G.f.: x*U(x)*(1 + U(x))/(1-x), where 1 + U(x) is the g.f. of A002212 (using the notation in the two papers by Cyvin et al. published in 1994).

More terms from Emeric Deutsch, Oct 24 2002

cp's OEIS Frontend

A104259 Triangle T read by rows: matrix product of Pascal and Catalan triangle.

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A026106 Number of polyhexes of class PF2 (with one catafusene annealated to pyrene).

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A026298 Number of polyhexes of class PF2.

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A030519 Number of polyhexes of class PF2 with four catafusenes annealated to pyrene.

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A182401 Number of paths from (0,0) to (n,0), never going below the x-axis, using steps U=(1,1), H=(1,0) and D=(1,-1), where the H steps come in five colors.

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A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

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A055450 Path-counting array T; each step of a path is (1 right) or (1 up) to a point below line y=x, else (1 right and 1 up) or (1 up) to a point on the line y=x, else (1 left) or (1 up) to a point above line y=x. T(i,j)=number of paths to point (i-j,j), for 1<=j<=i, i >= 1.

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A360100 a(n) = Sum_{k=0..n} binomial(n+2k-1,n-k) Catalan(k).