Fouad IBN MAJDOUB HASSANI - cp's OEIS frontend

User: Fouad IBN MAJDOUB HASSANI

Fouad IBN MAJDOUB HASSANI has authored 7 sequences.

A082398 Number of directed, diagonally convex polyominoes with n cells.

1, 0, 3, 10, 33, 106, 331, 1009, 3017, 8884, 25841, 74416, 212533, 602785, 1699503, 4767166, 13312641, 37031254, 102651967, 283676689, 781763381, 2149017256, 5894114513, 16132400860, 44071485673, 120188174401, 327242994651

Offset: 1

Fouad IBN MAJDOUB HASSANI, Apr 14 2003

Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux decales oscillants. These de Doctorat, Laboratoire de Recherche en Informatique, Universite Paris-Sud XI, France.

Join[{1, 0}, Take[CoefficientList[Series[(q*(1 - 4*q + 6*q^2 - 3*q^3 + q^4))/((1 - q)*(1 - 3*q + q^2)^2), {q, 0, 27}], q], -26]]

G.f.: (x*(1-4*x+6*x^2-3*x^3+x^4))/((1-x)*(1-3*x+x^2)^2)

More terms from Robert G. Wilson v, Apr 15 2003

A082397 Number of directed aggregates of height <= 2 with n cells.

Original entry on oeis.org

1, 2, 5, 11, 26, 62, 153, 385, 988, 2573, 6786, 18084, 48621, 131718, 359193, 985185, 2715972, 7521567, 20915256, 58373586, 163462815, 459136809, 1293223230, 3651864606, 10336625731, 29321683082, 83344398533, 237344961291

Offset: 1

Fouad IBN MAJDOUB HASSANI, Apr 14 2003

nonn

Conjecture: partial sums of A342912. - Sean A. Irvine, Jul 16 2022

Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux décalés oscillants. Thèse de Doctorat. 1991. Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.

Rigoberto Flórez and José L. Ramírez, Enumerating symmetric pyramids in Motzkin paths, Ars Math. Contemporanea (2023) Vol. 23, #P4.06.

A082397 := proc(n)
    add( (-1)^(k+1)*binomial(n+1,k+1)*binomial(k,floor((k-1)/2)),k=1..n) ;
end proc:
seq(A082397(n),n=1..30) ; # R. J. Mathar, Jun 27 2022

Table[Sum[(-1)^(i+1)*Binomial[k+1, i+1] Binomial[i, Floor[(i-1)/2]], {i,1,k}], {k,1,20}] (* Rigoberto Florez, Dec 10 2022 *)

import math
def Sum(k):
    S= sum((-1)**(i+1)*math.comb(k,i+1)*math.comb(i,math.floor((i-1)/2)) for i in range(1,k))
    return S
for i in range (2,20): print(Sum(i))
# Rigoberto Florez, Dec 10 2022

a(n) = Sum_{k=1..n}(-1)^(k+1)*binomial(n+1, k+1)*binomial(k, floor((k-1)/2)). E.g.f.: -exp(x)*int(-BesselI(1, 2*x)+BesselI(2, 2*x), x)-exp(x)*(-BesselI(1, 2*x)+BesselI(2, 2*x)). - Vladeta Jovovic, Sep 18 2003

Conjecture D-finite with recurrence +(n+2)*a(n) +(-3*n-2)*a(n-1) -n*a(n-2) +3*n*a(n-3)=0. - R. J. Mathar, Jun 27 2022

More terms from Vladeta Jovovic, Sep 18 2003

A082395 Number of shifted Young tableaux with height <= 3.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 27, 63, 154, 386, 989, 2574, 6787, 18085, 48622, 131719, 359194, 985186, 2715973, 7521568, 20915257, 58373587, 163462816, 459136810, 1293223231, 3651864607, 10336625732, 29321683083, 83344398534, 237344961292, 677087183363, 1934730432503

Offset: 1

Fouad IBN MAJDOUB HASSANI, Apr 14 2003

nonn

			G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 27*x^7 + 63*x^8 + 154*x^9 + 386*x^10 + 989*x^11 + 2574*x^12 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FouadIbn-Majdoub-Hassani, Combinatoire de polyominos et des tableaux décalés oscillants, Thèse de Doctorat, 1996, Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.

Partial sums of A005043.

Cf. A363555.

Table[Sum[(-1)^(k+1)*Binomial[n, k]*Binomial[k-1, Floor[k/2]],{k,1,n}],{n,1,20}]
RecurrenceTable[{3*(n-2)*a[n-3]+(2-n)*a[n-2]+(4-3n)*a[n-1]+n*a[n]==0,a[1]==1,a[2]==1,a[3]==2},a,{n,20}] (* Vaclav Kotesovec, Oct 02 2012 *)

def A082395():
    a, b, s, n = 1, 0, 1, 1
    yield a
    while True:
        s += b
        yield s
        n += 1
        a, b = b, (2*b+3*a)*(n-1)/(n+1)
A082395_list = A082395()
[next(A082395_list) for i in range(30)] # Peter Luschny, Sep 24 2014

a(n) = Sum_{k=1..n}(-1)^(k+1)*binomial(n, k)*binomial(k-1, floor(k/2)). - Vladeta Jovovic, Sep 18 2003
Recurrence: 3*(n-2)*a(n-3)+(2-n)*a(n-2)+(4-3*n)*a(n-1)+n*a(n)=0. - Vaclav Kotesovec, Oct 02 2012
Asymptotic: a(n) ~ 3^(n+3/2)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 02 2012
From Paul D. Hanna, Jul 08 2023: (Start)
G.f. A(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*(1-x^2)).
G.f. A(x) = A(x)^2 + (1 - A(x)^2)*x + (A(x) - A(x)^2)*x^2 + A(x)^2*x^3.
G.f. A(x) satisfies 0 = Sum_{n>=0} (-1)^n * x^(n*(n-3)/2) * A(x)^n / Product_{k=0..n+1} (1 - x^k*A(x)). (End)

More terms from Vladeta Jovovic, Sep 18 2003

A053091 F^3-convex polyominoes on the honeycomb lattice by number of cells.

Original entry on oeis.org

1, 3, 5, 6, 9, 11, 10, 15, 18, 14, 21, 23, 18, 30, 29, 21, 33, 35, 31, 39, 41, 30, 42, 54, 35, 51, 53, 38, 66, 54, 42, 63, 65, 60, 69, 70, 43, 75, 90, 54, 81, 83, 63, 93, 89, 62, 90, 95, 84, 99, 90, 77, 105, 126, 74, 111, 113, 60, 138, 119, 91, 126, 125, 108

Offset: 1

Fouad IBN MAJDOUB HASSANI, Feb 28 2000

nonn

The polyominoes are counted up to translations but not rotations and reflections. Thus, the unique domino with two cells is counted three times for its three orientations. - Michael Somos, Jun 21 2012

			x + 3*x^2 + 5*x^3 + 6*x^4 + 9*x^5 + 11*x^6 + 10*x^7 + 15*x^8 + 18*x^9 + ...
+---+
| o | a(1) = 1
+---------------+
| o o | o  |  o | a(2) = 3
|     |  o | o  |
+-------------------------------+
|  o  | o o |       | o   |   o |
| o o |  o  | o o o |  o  |  o  | a(3) = 5
|     |     |       |   o | o   |
+-------------------------------------------+
|         | o    |    o |  o  |      |      |
| o o o o |  o   |   o  | o o |  o o | o o  | a(4) = 6
|         |   o  |  o   |  o  | o o  |  o o |
|         |    o | o    |     |      |      |
+-------------------------------------------+
- _Michael Somos_, Jun 21 2012

Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux decales oscillants. These de Doctorat. Laboratoire de Recherche en Informatique, Universite Paris-Sud XI, France.
Alain Denise, Christoph Durr and Fouad Ibn-Majdoub-Hassani. Enumeration et generation aleatoire de polyominos convexes en reseau hexagonal (French) [enumeration and random generation of convex polyominoes in the honeycomb lattice]. In Proceedings of 9th Conference on Formal Power Series and Algebraic Combinatorics, pages 222-234, 1997.

Alain Denise, Christoph Duerr and Fouad Ibn-Majdoub-Hassani Enumeration et generation aleatoire de polyominos convexes en reseau hexagonal (French)

{a(n) = local(m = 4*n); if( n<1, 0, (-1)^n / 2 * polcoeff( sum( k=1, m, k * kronecker( 2, k) * if( k%4 == 3, x^k, x^(3*k)) / (1 + x^(4*k)), O(x^m)), m - 1))} /* Michael Somos, Jun 20 2012 */

{a(n) = if( n<1, 0, polcoeff( sum( i=1, n, x^i * (1 + x^i) / (1 - x^i) * ( sum( k=1, i, x^((i - k) * (i + k - 1)/2), x * O(x^(n - i))))^2 ), n))} /* Michael Somos, Jun 21 2012 */

Expansion of F^3(1, 1, q, 1) in powers of q where F^3(x, y, q, t) is the generating function defined in the FPSAC97 article. - Michael Somos, Jun 20 2012
G.f.: sum_{n >= 1} sum{d|n} b_d^2 * x^d * (1 + sign(n-d)), where b_0 = 0 and
b_i = x^binomial(i, 2) * sum_{k=1}^{i} x^(-binomial(i, 2)) for i >= 1 [corrected by Michael Somos, Jun 21 2012]

A053022 Number of subdiagonal directed diagonally-convex animals with given diagonal semiperimeter.

Original entry on oeis.org

1, 1, 3, 9, 30, 107, 396, 1506, 5848, 23087, 92381, 373830, 1527192, 6290054, 26090780, 108895558, 456992576, 1927176941, 8162557837, 34708427777, 148110582311, 634074235584, 2722547715302, 11721572763762, 50591504811439

Offset: 2

Fouad IBN MAJDOUB HASSANI, Feb 24 2000

nonn

F. Ibn-Majdoub-Hassani, Generating functions for directed animals convex following their direction, In: Actes du Seminaire Lotharingien de Combinatoire, 33rd session. Publication de l'IRMA, Strasbourg, France, 1994.
F. Ibn-Majdoub-Hassani, Combinatorics of polyominoes and oscillating shifted tableaux. Phd Thesis. Laboratoire de Recherche en Informatique, Univ. Paris-Sud, France, 1996

Seminaire Lotharingien de Combinatoire, Home page

G.f.: A(x)^4 + 2x(1 - x + x^2)A(x)^3 + x( - 1 + 3x - 7x^2 + 5x^3 - 2x^4 + x^5)A(x)^2 + x^3(2 - 5x + 6x^2 - 3x^3 + 2x^4)A(x) + x^5( - 1 + 2x - x^2 + x^3) = 0.

More terms from Vladeta Jovovic, Mar 27 2001

A053090 Number of F^3-convex polyominoes on honeycomb lattice with given semiperimeter.

Original entry on oeis.org

1, 0, 3, 2, 6, 6, 12, 12, 21, 22, 33, 36, 50, 54, 72, 78, 99, 108, 133, 144, 174, 188, 222, 240, 279, 300, 345, 370, 420, 450, 506, 540, 603, 642, 711, 756, 832, 882, 966, 1022, 1113, 1176, 1275, 1344, 1452, 1528, 1644, 1728, 1853, 1944, 2079, 2178, 2322, 2430

Offset: 3

Fouad IBN MAJDOUB HASSANI, Feb 28 2000

Sequence is also given by the Poincaré series [or Poincare series] of an ordinal Hodge algebra, or algebra with straightening law, that the three-strand braid group acts on. - Stephen P. Humphries, Feb 06 2009
From Michael Somos, Jun 21 2012: (Start)
Euler transform of length-6 sequence [ 0, 3, 2, 0, 0, -1].
Expansion of F^3(x, 1, 1, 1) in powers of x where F^3(x, y, q, t) is the generating function defined in the FPSAC97 article.
The polyominoes are counted up to translations but not rotations and reflections. Thus, the unique domino with two cells is counted three times for its three orientations.
The semiperimeter of each hexagonal cell is 3 but each common side shared by two cells decreases the semiperimeter by one. (End)
From Jianing Song, Apr 27 2022: (Start)
Let w = exp(2*Pi*i/6) be a primitive 6th root of 1. Then a(n+3) is number of nonnegative integer solutions to Sum_{j=0..5} x_j = n, Sum_{j=0..5} (x_j)*w^j = 0.
Proof: Sum_{j=0..5} (x_j)*w^j = x_0 + (x_1)*w + (x_2)*(w-1) - x_3 - (x_4)*w - (x_5)*(w-1), so Sum_{j=0..5} (x_j)*w^j = 0 if and only if x_0 + x_5 = x_2 + x_3 and x_1 + x_2 = x_4 + x_5, or equivalently, x_0 - x_3 = x_2 - x_5 = x_4 - x_1.
Case (a): x_0 - x_3 = x_2 - x_5 = x_4 - x_1 >= 0, then we can write x_0 = x+t, x_1 = z, x_2 = y+t, x_3 = x, x_4 = z+t, x_5 = y. The number of solutions in this case is equal to the number of solutions to 2*s + 3*t = n, x+y+z = s.
Case (b): x_0 - x_3 = x_2 - x_5 = x_4 - x_1 <= 0, then we can write x_0 = x, x_1 = z+t, x_2 = y, x_3 = x+t, x_4 = z, x_5 = y+t. The number of solutions in this case is also equal to the number of solutions to 2*s + 3*t = n, x+y+z = s.
The common part of case (a) and case (b) is the case where x_0 - x_3 = x_2 - x_5 = x_4 - x_1 = 0, in which case the number of solutions is equal to the number of solutions to x+y+z = n/2 for even n and 0 for odd n.
In conclusion, the total number of solutions is 2*Sum_{s=0..n/2, 3|(n-2*s)} (s+1)*(s+2)/2 - [n even]*(n/2+1)*(n/2+2)/2, where [] is an Iverson bracket. This can be shown to be equal to a(n+3). (End)

			x^3 + 3*x^5 + 2*x^6 + 6*x^7 + 6*x^8 + 12*x^9 + 12*x^10 + 21*x^11 + ...
+---+
| o | a(3) = 1
+---------------+
| o o | o  |  o | a(5) = 3
|     |  o | o  |
+---------------+
|  o  | o o | a(6) = 2
| o o |  o  |
+---------------------------------------+
|       | o   |   o |  o  |      | o o  |
| o o o |  o  |  o  | o o |  o o |  o o | a(7) = 6
|       |   o | o   |  o  | o o  |      |
+---------------------------------------+
- _Michael Somos_, Jun 21 2012

Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux décalés oscillants. Thèse de Doctorat. Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.
Alain Denise, Christoph Durr and Fouad Ibn-Majdoub-Hassani. Enumération et génération aléatoire de polyominos convexes en réseau hexagonal (French) [enumeration and random generation of convex polyominoes in the honeycomb lattice]. In Proceedings of 9th Conference on Formal Power Series and Algebraic Combinatorics (FPSAC97), pages 222-234, 1997.

Jianing Song, Table of n, a(n) for n = 3..10003
Alain Denise, Christoph Duerr and Fouad Ibn-Majdoub-Hassani, Enumération et génération aléatoire de polyominos convexes en réseau hexagonal (French)
Stephen P. Humphries, Action of some braid groups on Hodge algebras Comm. Algebra 26 (1998), no. 4, pages 1233-1242. See Proposition 3.4 on page 1241. [From _Stephen P. Humphries_, Feb 06 2009]
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1)

A053090 := proc(n)
    n*(2*n^2+3-27*(-1)^n)-32*A049347(n-1) ;
    %/144 ;
end proc:
seq(A053090(n),n=3..30) ; # R. J. Mathar, Mar 12 2025

{a(n) = round( n * (2*n^2 + 3) / 144 - (-1)^n * 3*n / 16)} /* Michael Somos, Jun 21 2012 */

{a(n) = sign(n) * polcoeff( x^3 * (1 + x^3) / ((1 - x^2)^3 * (1 - x^3)) + x * O(x^abs(n)), abs(n))} /* Michael Somos, Jun 21 2012 */

G.f.: x^3*(1 + x^3)/((1 - x^2)^3*(1 - x^3)).

a(-n) = -a(n). a(n) = round( n*(2*n^2 + 3)/144 - (-1)^n*3*n/16 ). - Michael Somos, Jun 21 2012

A053151 Number of directed EG-convex polyominoes on the honeycomb lattice with given semiperimeter.

Original entry on oeis.org

1, 0, 3, 2, 9, 12, 31, 53, 116, 215, 446, 849, 1726, 3320, 6688, 12932, 25925, 50297, 100546, 195562, 390257, 760630, 1516233, 2960502, 5897425, 11533047, 22964739, 44972711, 89528901, 175546608, 349425044, 685913758, 1365249931, 2682660933

Offset: 3

Fouad IBN MAJDOUB HASSANI, Feb 28 2000

Fouad Ibn-Majdoub-Hassani, Combinatoire de polyominos et des tableaux décalés oscillants, Thèse de Doctorat. Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.
Alain Denise, Christoph Durr, and Fouad Ibn-Majdoub-Hassani, Enumération et génération aléatoire de polyominos convexes en réseau hexagonal (French) [enumeration and random generation of convex polyominoes in the honeycomb lattice]. In Proceedings of 9th Conference on Formal Power Series and Algebraic Combinatorics, pages 222-234, 1997.

CoefficientList[Series[x^3*((1-2*x)*(-3+x+3*x^2+2*x^3)+(1-x-x^2)*(1-4*x^2)^(1/2))/(2*(1+x)*(1-2*x)*(1-x-x^2)*(-1+x+2*x^2+x^3)),{x,0,36}],x] (* Stefano Spezia, Oct 28 2023 *)

G.f.: x^3*((1-2*x)*(-3+x+3*x^2+2*x^3)+(1-x-x^2)*(1-4*x^2)^(1/2))/(2*(1+x)*(1-2*x)*(1-x-x^2)*(-1+x+2*x^2+x^3)).

More terms from James Sellers, Mar 01 2000

cp's OEIS Frontend

User: Fouad IBN MAJDOUB HASSANI

A082398 Number of directed, diagonally convex polyominoes with n cells.

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A082397 Number of directed aggregates of height <= 2 with n cells.

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A082395 Number of shifted Young tableaux with height <= 3.

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A053091 F^3-convex polyominoes on the honeycomb lattice by number of cells.

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PARI

PARI

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A053022 Number of subdiagonal directed diagonally-convex animals with given diagonal semiperimeter.

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A053090 Number of F^3-convex polyominoes on honeycomb lattice with given semiperimeter.

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Programs

Maple

PARI

PARI

Formula

A053151 Number of directed EG-convex polyominoes on the honeycomb lattice with given semiperimeter.

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Author

Keywords

References

Programs

Mathematica

Formula

Extensions