A003480 -id:A003480 - cp's OEIS frontend

A067311 Triangle read by rows: T(n,k) gives number of ways of arranging n chords on a circle with k simple intersections (i.e., no intersections with 3 or more chords) - positive values only.

1, 1, 2, 1, 5, 6, 3, 1, 14, 28, 28, 20, 10, 4, 1, 42, 120, 180, 195, 165, 117, 70, 35, 15, 5, 1, 132, 495, 990, 1430, 1650, 1617, 1386, 1056, 726, 451, 252, 126, 56, 21, 6, 1, 429, 2002, 5005, 9009, 13013, 16016, 17381, 16991, 15197, 12558, 9646, 6916, 4641, 2912, 1703, 924, 462, 210, 84, 28, 7, 1

Offset: 0

Henry Bottomley, Jan 14 2002

Row n contains 1 + n(n-1)/2 entries. - Emeric Deutsch, Jun 03 2009
Row sums are A001147 (double factorials).
Columns include A000108 (Catalan) for k=0 and A002694 for k=1.
Coefficients of Touchard-Riordan polynomials defined on page 3 of the Chakravarty and Kodama paper, related to the array A039599 through the polynomial numerators of Eqn. 2.1. - Tom Copeland, May 26 2016

			Rows start:
   1;
   1;
   2,   1;
   5,   6,   3,   1;
  14,  28,  28,  20,  10,   4,   1;
  42, 120, 180, 195, 165, 117,  70,  35,  15,   5,   1;
etc.,
i.e., there are 5 ways of arranging 3 chords with no intersections, 6 with one, 3 with two and 1 with three.

P. Flajolet and M. Noy, Analytic combinatorics of chord diagrams; in Formal Power Series and Algebraic Combinatorics, pp. 191-201, Springer, 2000.

Andrew Howroyd, Table of n, a(n) for n = 0..4525 (rows 0..30)
Alt.Math, alt.math.recreational discussion
H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
S. Chakravarty and Y. Kodama, A generating function for the N-soliton solutions of the Kadomtsev-Petviashvili II equation, arXiv preprint arXiv:0802.0524v2 [nlin.SI], 2008.
FindStat - Combinatorial Statistic Finder, The number of nestings of a perfect matching., The number of crossings of a perfect matching.
P. Flajolet and M. Noy, Analytic combinatorics of chord diagrams, [Research Report] RR-3914, INRIA. 2000.
P. Luschny, First 10 rows of triangle (taken from Luschny link below)
P. Luschny, Variants of Variations.
J.-G. Penaud, Une preuve bijective d'une formule de Touchard-Riordan, Discrete Math., 139, 1995, 347-360. [From _Emeric Deutsch_, Jun 03 2009]
H. Prodinger, On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs , arXiv:1102.5186 [math.CO], 2011.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Lucas Sá and Antonio M. García-García, The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos, arXiv:2104.07647 [hep-th], 2021.

A067310 has a different view of the same table.

Cf. A039599.

p := proc (n) options operator, arrow: sort(simplify((sum((-1)^j*q^((1/2)*j*(j-1))*binomial(2*n, n+j), j = -n .. n))/(1-q)^n)) end proc; for n from 0 to 7 do seq(coeff(p(n), q, i), i = 0 .. (1/2)*n*(n-1)) end do; # yields sequence in triangular form; Emeric Deutsch, Jun 03 2009

nmax = 15; se[n_] := se[n] = Series[ Sum[(-1)^j*q^(j(j-1)/2)*Binomial[2 n, n+j], {j, -n, n}]/(1-q)^n , {q, 0, nmax}];
t[n_, k_] := Coefficient[se[n], q^k]; t[n_, 0] = Binomial[2 n, n]/(n + 1);
Select[Flatten[Table[t[n, k], {n, 0, nmax}, {k, 0, 2nmax}] ], Positive] [[1 ;; 55]]
(* Jean-François Alcover, Jun 22 2011, after Emeric Deutsch *)

M(n)=1/(1-q)^n*sum(k=0, n, (-1)^k * ( binomial(2*n,n-k)-binomial(2*n,n-k-1)) * q^(k*(k+1)/2) );
for (n=0,10, print( Vec(polrecip(M(n))) ) ); /* print rows */
/* Joerg Arndt, Oct 01 2012 */

T(n,k) = Sum_{j=0..n-1} (-1)^j * C((n-j)*(n-j+1)/2-1-k, n-1) * (C(2n, j) - C(2n, j-1)).
Generating polynomial of row n is (1-q)^(-n)*Sum_{j=-n..n} (-1)^j*q^(j*(j-1)/2)*binomial(2*n,n+j). - Emeric Deutsch, Jun 03 2009
O.g.f. as a continued fraction: 1/(1 - t/(1 - (1 + x)*t/(1 - (1 + x + x^2)*t/(1 - (1 + x + x^2 + x^3)*t/(1 - ...))))) = 1 + t + (2 + x)*t^2 + (5 + 6*x +3*x^2 + x^4)*t^3 + .... See Chakravarty and Kodama, equation 3.8. - Peter Bala, Jun 13 2019

a(55) onwards from Andrew Howroyd, Nov 22 2024

A145839 Number of 3-compositions of n.

Original entry on oeis.org

1, 3, 15, 73, 354, 1716, 8318, 40320, 195444, 947380, 4592256, 22260144, 107902088, 523036176, 2535324816, 12289536016, 59571339552, 288761470848, 1399719859808, 6784893012864, 32888561860032, 159421452802624, 772767131681280, 3745851196992000

Offset: 0

Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008

A 3-composition of n is a matrix with three rows, such that each column has at least one nonzero element and whose elements sum up to n.
Matrix inverse of (A000217(A004736)*A154990). - Mats Granvik, Jan 19 2009
(1 +3*x +15*x^2 +73*x^3 + ...) = 1/(1 -3*x -6*x^2 -10*x^3 -15*x^4 - ...). - Gary W. Adamson, Jul 27 2009
For n>1, a(n) is the number of generalized compositions of n-1 when there are i^2/2 +3i/2 +1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Emanuele Munarini, Maddalena Poneti, and Simone Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8.
Index entries for linear recurrences with constant coefficients, signature (6,-6,2).

Cf. A003480 (2-compositions), A145840 (4-compositions), A145841 (5-compositions).

Column k=3 of A261780.

I:=[3,15,73]; [1] cat [n le 3 select I[n] else 6*Self(n-1) - 6*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Mar 07 2021

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+2, 2), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 01 2015

Table[Sum[Binomial[n+3*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Dec 31 2013 *)
a[n_]:= a[n]= If[n==0, 1, Sum[Binomial[n-j+2, 2]*a[j], {j,0,n-1}]]; Table[a[n], {n, 0, 20}] (* G. C. Greubel, Mar 07 2021 *)

@CachedFunction
def a(n):
    if n==0: return 1
    else: return sum( binomial(n-j+2,2)*a(j) for j in (0..n-1))
[a(n) for n in (0..25)] # G. C. Greubel, Mar 07 2021

a(n+3) = 6*a(n+2) - 6*a(n+1) + 2*a(n).
G.f.: (1-x)^3/(2*(1-x)^3 - 1).
a(n) = Sum_{k>=0} C(n+3*k-1,n) / 2^(k+1). - Vaclav Kotesovec, Dec 31 2013
a(n) = Sum_{j=0..n-1} binomial(n-j+2, 2)*a(j) with a(0) = 1. - G. C. Greubel, Mar 07 2021

Offset corrected by Alois P. Heinz, Aug 31 2015

A145840 Number of 4-compositions of n.

Original entry on oeis.org

1, 4, 26, 164, 1031, 6480, 40728, 255984, 1608914, 10112368, 63558392, 399478064, 2510804924, 15780945024, 99186608832, 623409013632, 3918258753416, 24627092844352, 154786536605216, 972866430709568, 6114673231661936, 38432026791933696, 241553493927992448

Offset: 0

Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008

A 4-composition of n is a matrix with four rows, such that each column has at least one nonzero element and whose elements sum up to n.

G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.
E. Munarini, M. Poneti and S. Rinaldi, Matrix compositions, Proceedings of Formal Power Series and Algebraic Combinatorics 2006, San Diego, USA, J. Remmel, M. Zabrocki (Editors) 445-456.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Milan Janjić, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
E. Munarini, M. Poneti, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8.
Index entries for linear recurrences with constant coefficients, signature (8,-12,8,-2).

Cf. A003480, A145839, A145841.

Column k=4 of A261780.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+3, 3), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 01 2015

Table[Sum[Binomial[n+4*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Dec 31 2013 *)

a(n+4) = 8*a(n+3)-12*a(n+2)+8*a(n+1)-2*a(n).
G.f.: (1-x)^4/(2*(1-x)^4-1).
a(n) = sum(k>=0, C(n+4*k-1,n) / 2^(k+1)). - Vaclav Kotesovec, Dec 31 2013

Offset corrected by Alois P. Heinz, Aug 31 2015

A145841 Number of 5-compositions of n.

Original entry on oeis.org

1, 5, 40, 310, 2395, 18501, 142920, 1104060, 8528890, 65885880, 508970002, 3931805460, 30373291380, 234634403620, 1812556389540, 14002041536004, 108166106338760, 835585763004880, 6454920038905520, 49864411953151840, 385203777033190008, 2975708406629602400

Offset: 0

Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008

A 5-composition of n is a matrix with five rows, such that each column has at least one nonzero element and whose elements sum up to n.

G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.
E. Munarini, M. Poneti and S. Rinaldi, Matrix compositions, Proceedings of Formal Power Series and Algebraic Combinatorics 2006, San Diego, USA, J. Remmel, M. Zabrocki (Editors) 445-456.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Milan Janjić, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
E. Munarini, M. Poneti, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8.
Index entries for linear recurrences with constant coefficients, signature (10,-20,20,-10,2).

Cf. A003480 (2-compositions), A145839 (3-compositions), A145840 (4-compositions).

Column k=5 of A261780.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+4, 4), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 01 2015

Table[Sum[Binomial[n+5*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Dec 31 2013 *)

a(n+5) = 10*a(n+4)-20*a(n+3)+20*a(n+2)-10*a(n+1)+2*a(n).
G.f.: (1-x)^5/(2*(1-x)^5-1).
a(n) = sum(k>=0, C(n+5*k-1,n) / 2^(k+1)). - Vaclav Kotesovec, Dec 31 2013

Offset changed from 1 to 0 by Alois P. Heinz, Aug 31 2015

A020727 Pisot sequence P(2,7): a(0)=2, a(1)=7, thereafter a(n+1) is the nearest integer to a(n)^2/a(n-1).

Original entry on oeis.org

2, 7, 24, 82, 280, 956, 3264, 11144, 38048, 129904, 443520, 1514272, 5170048, 17651648, 60266496, 205762688, 702517760, 2398545664, 8189147136, 27959497216, 95459694592, 325919783936, 1112759746560, 3799199418368, 12971278180352, 44286713884672

Offset: 0

David W. Wilson

nonn

Also Pisot sequence T(2,7). - R. K. Guy
It appears that a(n) = 4*a(n-1) - 2*a(n-2) (holds at least up to n = 1000 but is not known to hold in general).
The recurrence holds up to n = 10^5. - Ralf Stephan, Sep 03 2013
Empirical g.f.: (2-x)/(1-4*x+2*x^2). - Colin Barker, Feb 21 2012

Colin Barker, Table of n, a(n) for n = 0..1000

It appears that this is a subsequence of A003480.

See A008776 for definitions of Pisot sequences.

Iv:=[2,7]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..30]]; // Bruno Berselli, Feb 04 2016

RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 04 2016 *)

pisotP(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]-1/2));
  a
}
pisotP(50, 2, 7) \\ Colin Barker, Aug 08 2016

Edited by N. J. A. Sloane, Aug 17 2009 at the suggestion of R. J. Mathar.

A181327 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an even sum (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 2, 4, 3, 12, 12, 32, 41, 9, 86, 140, 54, 232, 451, 246, 27, 624, 1416, 1008, 216, 1680, 4357, 3811, 1215, 81, 4522, 13192, 13692, 5832, 810, 12172, 39455, 47380, 25254, 5400, 243, 32764, 116820, 159296, 102024, 29700, 2916, 88192, 343029, 523549

Offset: 0

Emeric Deutsch, Oct 13 2010

A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.
Row n has 1+floor(n/2) entries.
The sum of entries in row n is A003480(n).
T(n,0) = A181329(n).
Sum(k*T(n,k), k>=0) = A181328(n).
For the statistic "number of column with an odd sum" see A181308.

			T(2,1) = 3 because we have (0 / 2), (1 / 1), and (2 / 0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
2;
4,    3;
12,  12;
32,  41,  9;
86, 140, 54;

Alois P. Heinz, Rows n = 0..200, flattened
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Cf. A003480, A181308, A181328, A181329.

G := (1-z^2)^2/(1-2*z-2*z^2+z^4-3*t*z^2+t*z^4): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
       expand(add(add(`if`(i=0 and j=0, 0, b(n-i-j)*
       `if`(irem(i+j,2)=0, x, 1)), i=0..n-j), j=0..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 16 2014

b[n_] := b[n] = If[n == 0, 1, Expand[Sum[Sum[If[i == 0 && j == 0, 0, b[n-i-j] * If[Mod[i+j, 2] == 0, x, 1]], {i, 0, n-j}], {j, 0, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

G.f.: G(t,z) = (1-z)^2*(1+z)^2/(1-2*z-2*z^2+z^4-3*t*z^2+t*z^4).
The g.f. of column k is z^{2k}*(1-z^2)^2*(3-z^2)^k/(1-2z-2z^2+z^4)^{k+1}
The g.f. H(t,s,z), where z marks size and t (s) marks number of columns with an odd (even) sum, is H=(1-z^2)^2/(1-2z^2+z^4-2tz-3sz^2+sz^4).

A322402 Triangle read by rows: The number of chord diagrams with n chords and k topologically connected components, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 4, 6, 5, 0, 27, 36, 28, 14, 0, 248, 310, 225, 120, 42, 0, 2830, 3396, 2332, 1210, 495, 132, 0, 38232, 44604, 29302, 14560, 6006, 2002, 429, 0, 593859, 678696, 430200, 204540, 81900, 28392, 8008, 1430, 0, 10401712, 11701926, 7204821, 3289296, 1263780, 431256, 129948, 31824, 4862

Offset: 0

R. J. Mathar, Dec 06 2018

If all subsets are allowed instead of just pairs (chords), we get A324173. The rightmost column is A000108 (see Riordan). - Gus Wiseman, Feb 27 2019

			From _Gus Wiseman_, Feb 27 2019: (Start)
Triangle begins:
  1
  0      1
  0      1      2
  0      4      6      5
  0     27     36     28     14
  0    248    310    225    120     42
  0   2830   3396   2332   1210    495    132
  0  38232  44604  29302  14560   6006   2002    429
  0 593859 678696 430200 204540  81900  28392   8008   1430
Row n = 3 counts the following chord diagrams (see link for pictures):
  {{1,3},{2,5},{4,6}}  {{1,2},{3,5},{4,6}}  {{1,2},{3,4},{5,6}}
  {{1,4},{2,5},{3,6}}  {{1,3},{2,4},{5,6}}  {{1,2},{3,6},{4,5}}
  {{1,4},{2,6},{3,5}}  {{1,3},{2,6},{4,5}}  {{1,4},{2,3},{5,6}}
  {{1,5},{2,4},{3,6}}  {{1,5},{2,3},{4,6}}  {{1,6},{2,3},{4,5}}
                       {{1,5},{2,6},{3,4}}  {{1,6},{2,5},{3,4}}
                       {{1,6},{2,4},{3,5}}
(End)

P. Flajolet and M. Noy, Analytic Combinatorics of Chord Diagrams, in: Formal power series and algebraic combinatorics (FPSAC '00) Moscow, 2000, p 191-201, eq (2)
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Gus Wiseman, Chords diagrams with 3 chords, organized by number of components.

Cf. A000699 (k = 1 column), A001147 (row sums), A000108 (diagonal), A002694 (subdiagonal k = n - 1).

Cf. A000096, A003436, A016098, A099947, A136653, A278990, A293157, A324173, A324323, A324327, A324328.

The g.f. satisfies g(z,w) = 1+w*A000699(w*g^2), where A000699(z) is the g.f. of A000699.

Offset changed to 0 by Gus Wiseman, Feb 27 2019

A116484 Expansion of (-1+3x)/(5x^2 + 1 - 2*x).

Original entry on oeis.org

-1, 1, 7, 9, -17, -79, -73, 249, 863, 481, -3353, -9111, -1457, 42641, 92567, -28071, -518977, -897599, 799687, 6087369, 8176303, -14084239, -69049993, -67678791, 209892383, 758178721, 466895527, -2857102551, -8048682737, -1811852719

Offset: 0

Creighton Dement, Feb 17 2006

Binomial transform of signed powers of 2: (-1, 2, 4, -8, -16, 32, 64, -128, -256, 512, 1024). Inverse binomial transform of (-1, 0, 8, 32, 64, 0, -512, -2048, -4096, 0, 32768, 131072, 262144, 0, -2097152, -8388608). Compare with A116483.

Floretion Algebra Multiplication Program, FAMP Code: 2basekforseq[A*B] with A = - .5'i + .5'j - .5i' + .5j' + 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj' and B = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' ; 1vesforseq = A000004

Harvey P. Dale, Table of n, a(n) for n = 0..1000
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,-5).

Cf. A000079, A006495, A116483.

CoefficientList[Series[(-1+3x)/(5x^2+1-2x),{x,0,40}],x] (* or *) LinearRecurrence[{2,-5},{-1,1},40] (* Harvey P. Dale, Jun 24 2013 *)

a(n) = 3*A045873(n) - A045873(n+1). - R. J. Mathar, Apr 23 2009
E.g.f.: exp(x)*(sin(2*x) - cos(2*x)). - Arkadiusz Wesolowski, Aug 31 2012
a(0)=-1, a(1)=1, a(n) = 2*a(n-1) - 5*a(n-2). - Harvey P. Dale, Jun 24 2013
a(n) = (1/2)*((-1 - i)*(1 + 2*i)^n - (1 - i)*(1 - 2*i)^n), n >= 0, where i=sqrt(-1). - Taras Goy, Apr 20 2019

A126764 Number of L-convex polyominoes with n cells, that is, convex polyominoes where any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientations of the letter L).

Original entry on oeis.org

1, 1, 2, 6, 15, 35, 76, 156, 310, 590, 1098, 1984, 3515, 6094, 10398, 17434, 28837, 47038, 75820, 120794, 190479, 297365, 460056, 705576, 1073473, 1620680, 2429352, 3616580, 5349359, 7863564, 11491946, 16700534, 24140606, 34716813, 49682700, 70766326, 100343410

Offset: 0

N. J. A. Sloane, based on email from Simone Rinaldi (rinaldi(AT)unisi.it), Feb 23 2007

nonn

This sequence counts fixed L-convex polyominoes. See crossrefs for the free case. - Allan C. Wechsler, Jan 27 2023

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Combinatorial Exploration: An algorithmic framework for enumeration, arXiv:2202.07715 [math.CO], 2022.
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
Anthony Guttmann and Vaclav Kotesovec, L-convex polyominoes and 201-avoiding ascent sequences, arXiv:2109.09928 [math.CO], 2021.

Cf. A000105, A003480, A126765.

See A360055 for the free case.

nmax = 50; f[k_, x_] := f[k, x] = (If[k == 0, 1, If[k == 1, 1 + 2*x - x^2, Normal[Series[2*f[k-1, x] - (1 - x^k)^2 * f[k-2, x], {x, 0, nmax}]]]]); CoefficientList[Series[1 + Sum[x^k * f[k-1, x]/((Product[(1 - x^j)^2, {j, 1, k-1}] * (1 - x^k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2021 *)

The reference gives a generating function.

Conjecture: a(n) ~ c * exp(Pi*sqrt(13*n/6)) / n^(3/2), where c = 13*sqrt(2) / 768. - Anthony Guttmann and Vaclav Kotesovec, Jun 09 2021

Definition corrected at the suggestion of Emeric Deutsch, Mar 03 2007

More terms from Vaclav Kotesovec, Jun 06 2021

A116483 Expansion of (1 + x) / (5x^2 - 2x + 1).

Original entry on oeis.org

1, 3, 1, -13, -31, 3, 161, 307, -191, -1917, -2879, 3827, 22049, 24963, -60319, -245453, -189311, 848643, 2643841, 1044467, -11130271, -27482877, 685601, 138785587, 274143169, -145641597, -1661999039, -2595790093, 3118415009, 19215780483

Offset: 0

Creighton Dement, Feb 17 2006

Binomial transform of signed powers of 2: (1, 2, -4, -8, 16, 32, -64, -128, ...).
Inverse binonomial transform of (1, 4, 8, 0, -64, -256, -512, 0, 4096, 16384, 32768, 0, -262144, -1048576, -2097152, 0, ...).
G.f.*(1-x)/(1+x) (i.e, convolution with 1,-2,2,-2,2,-2, ... ) yields A006495.
Floretion Algebra Multiplication Program, FAMP Code: 2ibaseforseq[A*B] with A = - .5'i + .5'j - .5i' + .5j' + 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj' and B = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' ;

Colin Barker, Table of n, a(n) for n = 0..1000
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,-5).

Cf. A000079, A006495, A116484.

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])+imag(v[k]));}
/* cf. A138749 */ /* Joerg Arndt, Jul 06 2011 */

Vec((1 + x) / (5*x^2 - 2*x + 1) + O(x^50)) \\ Colin Barker, Aug 25 2017

a(n) = 2*a(n-1) -5*a(n-2). - Paul Curtz, Apr 18 2011

a(n) = (1/2 + i/2)*((1 - 2*i)^n - i*(1 + 2*i)^n) where i=sqrt(-1). - Colin Barker, Aug 25 2017

cp's OEIS Frontend

A067311 Triangle read by rows: T(n,k) gives number of ways of arranging n chords on a circle with k simple intersections (i.e., no intersections with 3 or more chords) - positive values only.

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A145839 Number of 3-compositions of n.

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A145840 Number of 4-compositions of n.

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A145841 Number of 5-compositions of n.

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A020727 Pisot sequence P(2,7): a(0)=2, a(1)=7, thereafter a(n+1) is the nearest integer to a(n)^2/a(n-1).

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A181327 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with an even sum (0<=k<=floor(n/2)).

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A116484 Expansion of (-1+3x)/(5x^2 + 1 - 2*x).

A116483 Expansion of (1 + x) / (5x^2 - 2x + 1).