A004148 -id:A004148 - cp's OEIS frontend

A197601 G.f.: exp( Sum_{n>=1} [Sum_{k=0..2n} C(2n,k)^2 x^k] x^n/n ).

1, 1, 5, 14, 52, 187, 708, 2734, 10758, 43004, 174004, 711660, 2936564, 12211688, 51124185, 215299685, 911445413, 3876523626, 16556573129, 70980163570, 305343924258, 1317634326631, 5702146948069, 24741071869651, 107608326588838, 469073933764287

Offset: 0

Paul D. Hanna, Oct 20 2011

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 14*x^3 + 52*x^4 + 187*x^5 + 708*x^6 +...
The logarithm of the g.f. begins:
log(A(x)) = x + 9*x^2/2 + 28*x^3/3 + 121*x^4/4 + 496*x^5/5 + 2100*x^6/6 + 9017*x^7/7 + 38969*x^8/8 +...+ A198059(n)*x^n/n +...
and equals the sum of the series:
log(A(x)) = (1 + 2^2*x + x^2)*x
+ (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^2/2
+ (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^3/3
+ (1 + 8^2*x + 28^2*x^2 + 56^2*x^3 + 70^2*x^4 + 56^2*x^5 + 28^2*x^6 + 8^2*x^7 + x^8)*x^4/4
+ (1 + 10^2*x + 45^2*x^2 + 120^2*x^3 + 210^2*x^4 + 252^2*x^5 + 210^2*x^6 + 120^2*x^7 + 45^2*x^8 + 10^2*x^9 + x^10)*x^5/5 +...
which involves the squares of the coefficients in even powers of (1+x).
The logarithm of the g.f. can also be expressed as:
log(A(x)) = (1-x)^5*(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 +...)*x
+ (1-x)^9*(1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 +...)*x^2/2
+ (1-x)^13*(1 + 7^2*x + 28^2*x^2 + 84^2*x^3 + 210^2*x^4 +...)*x^3/3
+ (1-x)^17*(1 + 9^2*x + 45^2*x^2 + 165^2*x^3 + 495^2*x^4 +...)*x^4/4
+ (1-x)^21*(1 + 11^2*x + 66^2*x^2 + 286^2*x^3 + 1001^2*x^4 +...)*x^5/5 +...
which involves the squares of the coefficients in odd powers of 1/(1-x).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A198059 (log), A186236, A004148, A180717, A180718.

nmax = 30; CoefficientList[Series[Exp[Sum[Hypergeometric2F1[-2*k, -2*k, 1, x]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 29 2022 *)

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(2*m,k)^2 *x^k) *x^m/m)+x*O(x^n)), n)}

{a(n)=polcoeff(exp(sum(m=1, n, (1-x+x*O(x^n))^(4*m+1) *sum(k=0, n-m+1, binomial(2*m+k, k)^2 *x^k+x*O(x^n)) *x^m/m)+x*O(x^n)), n)}

G.f.: exp( Sum_{n>=1} (1-x)^(4*n+1) *[Sum_{k>=0} C(2*n+k,k)^2 *x^k] *x^n/n ).

Logarithmic derivative equals A198059.

A202845 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of odd length (n>=0, k>=0).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 2, 6, 4, 10, 3, 7, 16, 14, 11, 30, 40, 1, 17, 62, 90, 16, 28, 126, 184, 85, 49, 241, 384, 295, 9, 87, 444, 839, 808, 105, 152, 820, 1845, 1960, 594, 2, 262, 1547, 3938, 4581, 2331, 76, 453, 2957, 8134, 10731, 7326, 771, 794, 5636, 16529, 25110, 20204, 4529, 30

Offset: 0

Emeric Deutsch, Dec 26 2011

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
Sum of entries in row n is A004148 (the secondary structure numbers).
Sum(k*T(n,k), k>=0)=A202846(n).
T(n,0)=A023427(n).

			Row 5 is 2,6: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, ABVBA, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv; except for the first two, each has 1 stack of length 1.
Triangle starts:
1;
1;
1;
1,1;
1,3;
2,6;
4,10,3;
7,16,14;

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A004148, A202846, A202848, A202849,

f := (t*z^2+z^4)/(1-z^4): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form

G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (tz^2 + z^4)/(1-z^4).

The multivariate g.f. H(z, t[1], t[2], ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + (f/(1 + f))H(H-1), where f = t[1]z^2 + t[2]z^4 + t[3]z^6 + ... .

A202846 Number of stacks of odd length in all 2ndary structures of size n.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 16, 44, 113, 290, 749, 1930, 4966, 12776, 32870, 84577, 217665, 560328, 1442893, 3716837, 9577805, 24689612, 63667585, 164239124, 423824628, 1094065998, 2825169786, 7297681867, 18856458451, 48737762624, 126007604078, 325873570924, 842982118807

Offset: 0

Emeric Deutsch, Dec 26 2011

nonn

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.

Number of stacks of even length in all 2ndary structures of size n+2.

			a(5)=6: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; they have 0,1,1,1,1,1,1,0 stacks of odd length, respectively.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A004148, A171854, A202845, A023427, A202848, A202849

g := z^2*(1-z^2)*S*(S-1)/((1+z^2)*(1-z+z^2-2*z^2*S)): S := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);

a(n) = Sum(k*A202845(n,k), k>=0).
a(n) = Sum(k*A202848(n+2,k), k>=0).
a(n)+a(n-2) = A171854(n) (n>=2).
G.f.: g(z) = z^2*(1-z^2)^2*S(S - 1)/[(1+z^2)(1 - z + z^2 -2*z^2*S)], where S is defined by S = 1 + z*S + z^2*S(S-1) (the g.f. of the secondary structure numbers A004148).
Conjecture D-finite with recurrence +(n+2)*(13230*n^2-96611*n+147133)*a(n) +(-44206*n^3+292903*n^2-261197*n-341332)*a(n-1) +2*(17746*n^3-141629*n^2+231187*n+123600)*a(n-2) +2*(-26460*n^3+157889*n^2-64195*n-381418)*a(n-3) +2*(35492*n^3-320849*n^2+745453*n-240088)*a(n-4) +2*(-13230*n^3+98869*n^2-160610*n-79637)*a(n-5) +(48722*n^3-428591*n^2+982443*n-433110)*a(n-6) -(n-6)*(17746*n^2-68387*n+43705)*a(n-7)=0. - R. J. Mathar, Jul 26 2022

A202848 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of even length (n>=0, k>=0).

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 1, 14, 3, 31, 6, 66, 16, 141, 44, 313, 107, 3, 702, 262, 14, 1577, 663, 43, 3581, 1654, 138, 8207, 4091, 436, 1, 18903, 10178, 1275, 16, 43770, 25339, 3638, 85, 101903, 62952, 10316, 331, 238282, 156495, 28743, 1228, 559322, 389374, 78979, 4320, 9

Offset: 0

Emeric Deutsch, Dec 26 2011

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
Sum of entries in row n is A004148 (the secondary structure numbers).
Sum(k*T(n,k), k>=0) = A202846(n-2).
T(n,0) = A202849(n).

			Row 5 is 7,1: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv; ABVBA; the last one has 1 stack of length 2.
Triangle starts:
1;
1;
1;
2;
4;
7,1;
14,3;
31,6;

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A004145, A202846, A023427, A202849,

f := (z^2+t*z^4)/(1-z^4): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 19 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form

G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (z^2 + t*z^4)/(1-z^4).

The multivariate g.f. H(z, t[1], t[2], ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + (f/(1 + f))H(H-1), where f = t[1]z^2 + t[2]z^4 + t[3]z^6 + ... .

A292460 Expansion of (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4x^3))/(2x^3) in powers of x.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199, 134474581374

Offset: 0

Seiichi Manyama, Sep 16 2017

nonn

Number of U_{k}D-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. - Sergey Kirgizov, Apr 08 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.

Cf. A001006, A004148, A292461.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2 -Sqrt((1-x-x^2)^2 -4*x^3))/(2*x^3))); // G. C. Greubel, Aug 13 2018

CoefficientList[Series[(1-x-x^2 -Sqrt[(1-x-x^2)^2 -4*x^3])/(2*x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2018 *)

x='x+O('x^50); Vec((1-x-x^2 -sqrt((1-x-x^2)^2 -4*x^3))/(2*x^3)) \\ G. C. Greubel, Aug 13 2018

G.f.: 1/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(... (continued fraction).
a(n) = A004148(n+1).
a(n) ~ 5^(1/4) * phi^(2*n + 4) / (2*sqrt(Pi)*n^(3/2)), where phi is the golden ratio (1+sqrt(5))/2. - Vaclav Kotesovec, Sep 17 2017
D-finite with recurrence: (n+3)*a(n) +(-2*n-3)*a(n-1) -n*a(n-2) +(-2*n+3)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Jan 23 2020
a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-3} a(k) * a(n-k-3). - Ilya Gutkovskiy, Nov 09 2021

A365690 G.f. satisfies A(x) = 1 + x^2A(x)^4 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 1, 1, 5, 10, 38, 101, 353, 1070, 3659, 11843, 40505, 135873, 468104, 1604375, 5576315, 19386656, 67950717, 238676813, 842797959, 2983745508, 10603445402, 37777263153, 134985354179, 483438728094, 1735527037388, 6243193190117, 22503637842423

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A004148, A005043, A025246, A046736, A365691.

Cf. A365692.

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,n-2*k) * binomial(n+2*k+1,k) / (n+2*k+1).

A023425 Generalized Catalan numbers: a(0) = 1, a(n) = a(n-1) + Sum_{k=1..n-4} a(k) * a(n-k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 13, 34, 90, 241, 652, 1780, 4899, 13581, 37893, 106340, 299978, 850187, 2419788, 6913658, 19822439, 57015620, 164476023, 475752469, 1379553027, 4009532279, 11678165796, 34081307147, 99646051271, 291845778020, 856147139606, 2515368741707, 7400713869808, 21803597196231

Offset: 0

Olivier Gérard

a(n) = number of bargraphs of semiperimeter n-2 with no valleys of width 1 (i.e., no DHU configurations, where U=(0,1), H=(1,0), D=(0,-1)). Example: a(8) = 34 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only the one corresponding to the composition [2,1,2] has a valley of width 1. - Emeric Deutsch, Aug 11 2016

Cf. A000108, A001006, A004148, A006318, A082582.

A023425 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        procname(n-1)+add(procname(k)*procname(n-k),k=1..n-4) ;
    end if;
end proc: # R. J. Mathar, May 01 2015

a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-k ], {k, 1, n-4} ];

The sequence a(n-3) (for n>=3) has the g.f. 1/G(0) where G(k) = 1 - q/(1 - q - q^2 - q^3 / G(k+1) ). - Joerg Arndt, Dec 06 2014
n*a(n) +2*(-2*n+3)*a(n-1) +2*(n-3)*a(n-2) +(2*n-9)*a(n-3) +(n-6)*a(n-4) +(2*n-15)*a(n-5) +(n-9)*a(n-6)=0. - R. J. Mathar, May 01 2015
G.f.: (3+z^2+z^3-sqrt((1-2*z-3*z^2-z^3)*(1-2*z+z^2-z^3)))/2. - Emeric Deutsch, May 24 2016
The g.f. g(x) satisfies g(x) = 1+x*g(x)+(g(x)-1)*(g(x)-x^3-x^2-x-1) and 3*x^5+5*x^4+7*x^3+6*x^2+2*x-3+(-3*x^5-5*x^4-2*x^3-3*x^2-2*x+2)*g(x)+(x^6+2*x^5+x^4+2*x^3+2*x^2-4*x+1)*g'(x). - Robert Israel, May 25 2016

A045829 Catafusenes (see reference for precise definition).

Original entry on oeis.org

0, 0, 0, 1, 12, 94, 612, 3605, 19992, 106644, 554184, 2827902, 14244120, 71073860, 352180920, 1736103460, 8525167680, 41741310400, 203929367040, 994680578505, 4845761001756, 23586190895078, 114731538098100, 557859491227841

Offset: 1

N. J. A. Sloane

nonn

The sequence without the initial 0's is the 4-fold convolution of A002212(n), n = 1,2,... . - Emeric Deutsch, Mar 13 2004

The 2-fold convolution of A045445 (apart from zeros). - R. J. Mathar, Aug 01 2019

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; see Table 4 (p. 1177).
Asamoah Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.
Asamoah Nkwanta, Predicting RNA secondary structures: A lattice walk approach to modeling sequences within the HIV-1 RNA structure, slides of a talk given in Johannesburg, South Africa, 2006. [The slides may not necessarily contain this sequence, but they give the background for the above paper in the DIMACS book.]

Cf. A002212, A004148, A045445.

G.f.: (z*M)^4, where M = (1 - 3*z - sqrt(1-6*z+5*z^2))/(2*z^2). - Emeric Deutsch, Mar 13 2004

a(n) ~ 2 * 5^(n + 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, May 29 2022

More terms from Emeric Deutsch, Mar 13 2004

A097100 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n containing k subwords of the type U H^j U or D H^j D for some j>0, where U=(1,1), H=(1,0) and D=(1,-1) (can be easily expressed using RNA secondary structure terminology).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 2, 28, 8, 1, 53, 24, 5, 102, 62, 21, 199, 152, 68, 4, 391, 366, 196, 24, 1, 773, 868, 531, 104, 7, 1537, 2032, 1393, 368, 43, 3075, 4694, 3593, 1172, 195, 6, 6189, 10732, 9120, 3528, 754, 48, 1, 12525, 24348, 22822, 10224, 2632, 272, 9

Offset: 0

Emeric Deutsch, Sep 15 2004

Row sums are the RNA secondary structure numbers (A004148).
T(n,0)=A190160(n).
Sum(k*T(n,k),k>=0)=A190161(n).
The generating function G=G(t,s,z) relative to the number of subwords of the form uh^bu (marked by t) and dh^bd (marked by s) for a fixed b>=1, satisfies G = 1+zG+z^2*G[z/(1-z) + (w^2+twz^b+swz^b+tsz^{2b})H], where H=(1-z)[(1-z)G-1] and w = 1/(1-z) - z^b.

			Triangle starts:
  1;
  1;
  1;
  2;
  4;
  8;
  15,2;
  28,8,1;
  53,24,5;
  ...
It seems that, except for the first 3 rows, rows 4n-1, 4n, 4n+1 have 2n-1 terms and rows 4n+2 have 2n terms (n=1,2,...).
T(8,2)=5 because we have (UHU)H(DHD)H, (UHU)HH(DHD), H(UHU)H(DHD), (UHHU)H(DHD) and (UHU)H(DHHD); the required subwords are shown between parentheses.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08; Sem. Loth. Comb. B08l (1984) 79-86.

Cf. A004148, A190160, A190161.

eq := G = 1+z*G+z^2*G*(z+(1-z+t*z)^2*(G-z*G-1))/(1-z): G:= RootOf(eq,G): Gser := simplify(series(G,z=0,20)): for n from 0 to 19 do P[n] := sort(coeff(Gser,z,n)) end do: for n from 0 to 19 do seq(coeff(P[n],t,j), j=0 .. degree(P[n])) end do; # yields sequence in triangular form

G.f.: G=G(t, z) satisfies G=1+zG+z^2*G*[z+(1-z+t*z)^2*(G-zG-1)]/(1-z).

A097779 Number of Motzkin paths of length n, starting with an up step, ending with a down step and having no peaks (can be easily expressed using RNA secondary structure terminology).

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 5, 11, 25, 58, 135, 317, 750, 1785, 4272, 10275, 24823, 60210, 146576, 358010, 877087, 2154751, 5307166, 13102511, 32418806, 80375267, 199650310, 496803811, 1238276667, 3091173482, 7727893389, 19346109435, 48493869237

Offset: 0

Emeric Deutsch, Sep 11 2004

nonn

			a(6)=5 because we have UHHHHD, UHDUHD, UUHHDD, UHUHDD and UUHDHD, where U=(1,1), D=(1,-1) and H=(1,0).

Cf. A004148.

G:=z+1/2*(1-z)^2/z^2*(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4)): Gser:=series(G,z=0,40): 1,seq(coeff(Gser,z^n),n=1..37);

CoefficientList[Series[x+(1-x)^2 (1-x+x^2-Sqrt[1-2x-x^2-2x^3+x^4])/(2x^2),{x,0,40}],x] (* Harvey P. Dale, Dec 24 2016 *)

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

D-finite with recurrence (n+2)*a(n) -3*n*a(n-1) +(n-4)*a(n-2) +(-n+1)*a(n-3) +3*(n-5)*a(n-4) +(-n+7)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

A197601 G.f.: exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 *x^k] *x^n/n ).

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A202845 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of odd length (n>=0, k>=0).

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A202846 Number of stacks of odd length in all 2ndary structures of size n.

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A202848 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of even length (n>=0, k>=0).

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A292460 Expansion of (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4*x^3))/(2*x^3) in powers of x.

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A365690 G.f. satisfies A(x) = 1 + x^2*A(x)^4 / (1 - x*A(x)).

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A023425 Generalized Catalan numbers: a(0) = 1, a(n) = a(n-1) + Sum_{k=1..n-4} a(k) * a(n-k).

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A045829 Catafusenes (see reference for precise definition).

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A197601 G.f.: exp( Sum_{n>=1} [Sum_{k=0..2n} C(2n,k)^2 x^k] x^n/n ).

A292460 Expansion of (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4x^3))/(2x^3) in powers of x.

A365690 G.f. satisfies A(x) = 1 + x^2A(x)^4 / (1 - xA(x)).