A093128 -id:A093128 - cp's OEIS frontend

A132461 Row squared sums of triangle of Lucas polynomials (A034807) for n>0: Sum_{k=0..floor(n/2)} A034807(n,k)^2, with a(0)=1.

1, 1, 5, 10, 21, 51, 122, 295, 725, 1792, 4455, 11133, 27930, 70305, 177483, 449160, 1139157, 2894625, 7367720, 18781387, 47941271, 122524216, 313484385, 802877055, 2058184346, 5280670051, 13559216117, 34841384560, 89587774395

Offset: 0

Paul D. Hanna, Aug 21 2007

nonn

Also equals row squared sums of triangle A132460 and so equals the sum of the initial floor(n/2)+1 squared terms of 1/C(x)^n where C(x) is the g.f. of the Catalan numbers (A000108).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Marcos Mariño and Claudia Rella, On the structure of wave functions in complex Chern-Simons theory, arXiv:2312.00624 [hep-th], 2023. See p. 23.

Cf. A034807 (Lucas polynomials); A093128, A132460, A132459; A000108 (Catalan).

Flatten[{1,Table[Sum[(Binomial[n-k,k] + Binomial[n-k-1,k-1])^2,{k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 28 2014 *)

{a(n)=sum(k=0,n\2,(binomial(n-k, k)+binomial(n-k-1, k-1))^2)}
for(n=0, 30, print1(a(n), ", "))

/* squared sums of negative powers of Catalan series: */
{a(n)=local(Catalan=2/(1+sqrt(1-4*x +x*O(x^n)))); sum(k=0,n\2,polcoeff(Catalan^-n,k)^2)}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..floor(n/2)} ( C(n-k, k) + C(n-k-1, k-1) )^2.
Ignoring initial term, equals the logarithmic derivative of A093128, which gives the number of dissections of a polygon using strictly disjoint diagonals. - Paul D. Hanna, Nov 09 2013
From Vaclav Kotesovec, Feb 28 2014: (Start)
Recurrence (for n>=5): (n-3)*n*a(n) = (2*n^2 - 7*n + 4)*a(n-1) + (n-4)*n*a(n-2) + (2*n^2 - 9*n + 8)*a(n-3) - (n-4)*(n-1)*a(n-4).
G.f.: (2-x+2*x^2)/sqrt((x^2+x+1)*(x^2-3*x+1))-1.
a(n) ~ 5^(3/4) * ((3+sqrt(5))/2)^n / (2*sqrt(Pi*n)).
(End)

A171185 G.f.: exp( Sum_{n>=1} (x^n/n)*[Sum_{k=0..[n/2]} A034807(n,k)^3] ), where A034807 is a triangle of Lucas polynomials.

Original entry on oeis.org

1, 1, 5, 14, 40, 126, 408, 1332, 4473, 15377, 53627, 189724, 680475, 2467975, 9038578, 33399571, 124400702, 466619283, 1761467038, 6688059913, 25527326897, 97901917060, 377123873505, 1458573962761, 5662223702216, 22056563938599

Offset: 0

Paul D. Hanna, Dec 14 2009

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 14*x^3 + 40*x^4 + 126*x^5 + 408*x^6 +...
log(A(x)) = x + 9*x^2/2 + 28*x^3/3 + 73*x^4/4 + 251*x^5/5 + 954*x^6/6 +...+ A171215(n)*x^n/n +...

Cf. A171215, A093128, A171186.

{a(n)=polcoeff(exp(sum(m=1,n,(x^m/m)*sum(k=0, m\2, (binomial(m-k, k)+binomial(m-k-1, k-1))^3))+x*O(x^n)),n)}

A368376 Arises from enumeration of a certain class of zig-zag knight's paths on the square grid.

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 1, 6, 3, 13, 9, 29, 25, 65, 66, 148, 171, 341, 437, 793, 1107, 1860, 2790, 4395, 7009, 10452, 17574, 24999, 44019, 60097, 110210, 145130, 275925, 351916, 690993, 856502, 1731224, 2091599, 4339980, 5123437, 10887192, 12585354, 27331465

Offset: 0

N. J. A. Sloane, Feb 18 2024

nonn

It would be nice to have a more precise definition.

Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022). Also arXiv:2206.12087 [math.CO], Jan 2023. See Section 2.2.

Cf. A144366, A166297, A368377, A004148.

A093128 is a bisection.

r = (1 - z^4 - z^2 - Sqrt[z^8 - 2z^6 - z^4 - 2z^2 + 1]) / (2z^3);
gf = r (u^2 z + u z^2 + 1) / (z^3 (1 - r u));
Table[SeriesCoefficient[gf,{u,0,2},{z,0,n}], {n,0,33}] (* Andrei Zabolotskii, Jul 25 2025 *)

G.f.: (x + x^2 * R(x) + R(x)^2) * R(x) / x^3, where R(x) = x * (A(x^2) - 1) and A(x) is the g.f. of A004148. - Andrei Zabolotskii, Jul 25 2025

Terms a(14) and beyond from Andrei Zabolotskii, Jul 25 2025

A387202 a(n) is the number of dissections of a (4*n+2)-gon into hexagons using strictly disjoint diagonals.

Original entry on oeis.org

1, 5, 21, 87, 363, 1534, 6570, 28492, 124944, 553301, 2471373, 11122275, 50389695, 229643895, 1052093655, 4842863465, 22386911925, 103885321615, 483759492255, 2259888333445, 10587902977185, 49738841822400, 234235771140876, 1105609645231322, 5229610939919718

Offset: 1

Muhammed Sefa Saydam, Aug 21 2025

Muhammed Sefa Saydam, Table of n, a(n) for n = 1..100

Cf. A002212, A093128.

seq(n)={my(g=(1 - 3*x - sqrt(1 - 6*x + 5*x^2 + O(x*x^n)))/(2*x)); Vec((1 + 4*g + 3*g^2)*x + g^2)} \\ Andrew Howroyd, Aug 21 2025

G.f.: x*(1 + 4*B(x) + 3*B(x)^2) + B(x)^2, where 1 + B(x) is the g.f. of A002212. - Andrew Howroyd, Aug 21 2025

D-finite with recurrence -(n+2)*(2*n-3)*a(n) +3*(2*n+1)*(2*n-3)*a(n-1) -5*(2*n+1)*(n-3)*a(n-2)=0. - R. J. Mathar, Aug 28 2025

A387224 Number of dissections of a convex n-gon by strictly disjoint diagonals so as to create no triangles.

Original entry on oeis.org

0, 1, 1, 4, 8, 17, 37, 81, 177, 389, 859, 1905, 4241, 9477, 21251, 47806, 107864, 244045, 553575, 1258687, 2868285, 6549757, 14985361, 34347444, 78860152, 181347591, 417653187, 963234195, 2224464087, 5143567237, 11907471643, 27597112946, 64028244032, 148703128913, 345690623119

Offset: 3

Muhammed Sefa Saydam, Aug 22 2025

Strictly disjoint diagonals means that the diagonals are non-crossing and may not share endpoints.

			         n=4                         n=5                            n=6
    (1)       (2)                    (1)              (1) (2)     (1) (2)     (1) (2)
                                  (5)   (2)         (6)  \  (3) (6)-----(3) (6)  /  (3)
    (4)       (3)                  (4) (3)            (5) (4)     (5) (4)     (5) (4)
 Diagonal cannot be drawn   Diagonal cannot be drawn
    Number of cases = 1       Number of cases = 1         Number of cases = 3

Muhammed Sefa Saydam, Table of n, a(n) for n = 3..104

Cf. A004149, A093128.

seq(n) = my(g=2/(1 - x + x^2 + x^3 + sqrt((1-x^4)*(1-2*x-x^2) + O(x*x^n)))); Vec((1 - x^2 - 2*x^3)*g - 1 - x + 2*x^3 + 2*x^4, -n+2) \\ Andrew Howroyd, Aug 28 2025

a(n) = A004149(n) - A004149(n-2) - 2*A004149(n-3) for n >= 5.

G.f.: (1 - x^2 - 2*x^3)*B(x) - 1 - x + 2*x^3 + 2*x^4, where B(x) is the g.f. of A004149. - Andrew Howroyd, Aug 28 2025

A387267 Number of dissections of a convex n-gon into quadrilaterals and pentagons by strictly disjoint diagonals.

Original entry on oeis.org

0, 1, 1, 3, 7, 8, 19, 31, 47, 87, 135, 219, 371, 579, 947, 1535, 2423, 3919, 6239, 9891, 15803, 24987, 39563, 62663, 98751, 155815, 245431, 385771, 606467, 951795, 1492323, 2338703, 3660551, 5725951, 8950543, 13978931, 21820235, 34037067, 53059643, 82670167

Offset: 3

Muhammed Sefa Saydam, Aug 24 2025

Strictly disjoint diagonals means that the diagonals are non-crossing and may not share endpoints.

Muhammed Sefa Saydam, Table of n, a(n) for n = 3..100

Cf. A093128, A159284, A387224.

a(n) = T(n-3) + Sum_{i=1..n-8} T(i)*( T(n-i-4) + T(n-i-7) ) + Sum_{i=n-7..n-5} T(i)*( 1 + T(n-i-4) ) for n >= 9 and T(n) = A159284(n).

cp's OEIS Frontend

A132461 Row squared sums of triangle of Lucas polynomials (A034807) for n>0: Sum_{k=0..floor(n/2)} A034807(n,k)^2, with a(0)=1.

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A171185 G.f.: exp( Sum_{n>=1} (x^n/n)*[Sum_{k=0..[n/2]} A034807(n,k)^3] ), where A034807 is a triangle of Lucas polynomials.

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A368376 Arises from enumeration of a certain class of zig-zag knight's paths on the square grid.

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A387202 a(n) is the number of dissections of a (4*n+2)-gon into hexagons using strictly disjoint diagonals.

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A387224 Number of dissections of a convex n-gon by strictly disjoint diagonals so as to create no triangles.

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A387267 Number of dissections of a convex n-gon into quadrilaterals and pentagons by strictly disjoint diagonals.

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