A145841 -id:A145841 - cp's OEIS frontend

A003480 a(0) = 1, a(1) = 2, for n > 1, a(n) = 4a(n-1) - 2a(n-2).

1, 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, 151204299177984

Offset: 0

N. J. A. Sloane

Gives the 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 orientation of the L). - Simone Rinaldi (rinaldi(AT)unisi.it), Feb 19 2007
Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 2) is "size of raises in pot-limit poker, one blind, maximum raising".
Dimensions of the graded components of the Hopf algebra of noncommutative multi-symmetric functions of level 2. For level r, the sequence would be the INVERT transform of binomial(n+r-1,n). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
The sum of the numbers in the n-th row of the summatory Pascal triangle (A059576). - Ron R. King, Jan 22 2009
(1 + 2x + 7x^2 + 24x^3 + ...) = 1 / (1 - 2x - 3x^2 - 4x^3 - ...). - Gary W. Adamson, Jul 27 2009
Let M be a triangle with the odd-indexed Fibonacci numbers (1, 2, 5, 13, ...) in every column, with the leftmost column shifted upwards one row. A003480 = lim_{n->oo} M^n, the left-shifted vector considered as a sequence. The analogous operation using the even-indexed Fibonacci numbers generates A001835 starting with offset 1. - Gary W. Adamson, Jul 27 2010
a(n) is the number of generalized compositions of n when there are i+1 different types of the part i, (i=1,2,...). - Milan Janjic, Sep 24 2010
Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t + 3*t^2 + 4*t^3 + ...)),
  an o.g.f. for A003480, then
  A001003(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. - Tom Copeland, Sep 06 2011
Excluding the initial 1, a(n) is the 2nd subdiagonal of A228405. - Richard R. Forberg, Sep 02 2013

G. Castiglione and A. Restivo, L-convex polyominoes: a survey, Chapter 2 of K. G. Subranian et al., eds., Formal Models, Languages and Applications, World Scientific, 2015.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
Daniela Battaglino, Jean-Marc Fédou, Simone Rinaldi, and Samanta Socci, The number of k-parallelogram polyominoes, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1143-1154.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
Adrien Boussicault, Simone Rinaldi, and Samanta Socci, The number of directed k-convex polyominoes, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages. See t_n.
Steve Butler, Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo, Enumerating multiplex juggling patterns, arXiv:1702.05808 [math.CO], 2017.
Peter J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Giuseppa Castiglione, Andrea Frosini, Emanuele Munarini, Antonio Restivo, and Simone Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
Yumin Cho, Jaehyun Kim, Jang Soo Kim, and Nakyung Lee, Enumeration of multiplex juggling card sequences using generalized q-derivatives, arXiv:2402.09903 [math.CO], 2024. See p. 6.
Ana Djurdjevac, Vesa Kaarnioja, Claudia Schillings, and André-Alexander Zepernick, Uncertainty quantification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations, arXiv:2502.12345 [math.NA], 2025. See p. 34.
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
Enrica Duchi, Simone Rinaldi, and Gilles Schaeffer, The number of Z-convex polyominoes, arXiv:math/0602124 [math.CO], 2006.
Andrea Frosini and Simone Rinaldi, An object grammar for the class of L-convex polyominoes, PU.M.A. Vol. 17 (2006), No. 1-2, pp. 97-110.
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See pp. 12, 18
Yu-hong. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (12).
Harri Hakula, Helmut Harbrecht, Vesa Kaarnioja, Frances Y. Kuo, and Ian H. Sloan, Uncertainty quantification for random domains using periodic random variables, arXiv:2210.17329 [math.NA], 2022.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 418
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
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 sequences related to poker
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Row sums of A059576 and of A181289. Second differences of A007070.
Cf. A007052, A126764.
Cf. A000129, A001003, A001835, A006012, A029744, A145839, A145840, A145841, A228405.
Column k=2 of A261780.

a003480 n = a003480_list !! n
a003480_list = 1 : 2 : 7 : (tail $ zipWith (-)
   (tail $ map (* 4) a003480_list) (map (* 2) a003480_list))
-- Reinhard Zumkeller, Jan 16 2012, Oct 03 2011

INVERT([seq(n+1,n=1..20)]); # Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

a[0]=1; a[1]=2; a[2]=7; a[n_]:=a[n]=4*a[n-1] - 2*a[n-2]; Table[a[n],{n,0,24}] (* Jean-François Alcover, Mar 22 2011 *)
Join[{1},LinearRecurrence[{4,-2},{2,7},40]] (* Harvey P. Dale, Oct 23 2011 *)

a(n)=polcoeff((1-x)^2/(1-4*x+2*x^2)+x*O(x^n),n)

a(n)=local(x); if(n<1,n==0,x=(2+quadgen(8))^n; imag(x)+real(x)/2)

a(n) = (n+1)*a(0) + n*a(1) + ... + 3*a(n-2) + 2*a(n-1). - Amarnath Murthy, Aug 17 2002
G.f.: (1-x)^2/(1-4*x+2*x^2). - Simon Plouffe in his 1992 dissertation
a(n) = A007070(n)/2, n > 0.
G.f.: 1/( 1 - Sum_{k>=1} (k+1)*x^k ).
a(n+1)*a(n+1) - a(n+2)*a(n) = 2^n, n > 0. - D. G. Rogers, Jul 12 2004
For n > 0, a(n) = ((2+sqrt(2))^(n+1) - (2-sqrt(2))^(n+1))/(4*sqrt(2)). - Rolf Pleisch, Aug 03 2009
If the leading 1 is removed, 2, 7, 24, ... is the binomial transform of 2, 5, 12, 29, ..., which is A000129 without its first 2 terms, and the second binomial transform of 2, 3, 4, 6, ..., which is A029744, again without its leading 1. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
a(n) = Sum((1+p_1)*(1+p_2)*...*(1+p_m)), summation being over all compositions (p_1, p_2, ..., p_m) of n. Example: a(3)=24; indeed, the compositions of 3 are (1,1,1), (1,2), (2,1), (3) and we have 2*2*2 + 2*3 + 3*2 + 4 = 24. - Emeric Deutsch, Oct 17 2010
a(n) = Sum_{k>=0} binomial(n+2*k-1,n) / 2^(k+1). - Vaclav Kotesovec, Dec 31 2013
E.g.f.: (1 + exp(2*x)*(cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)))/2. - Stefano Spezia, May 20 2024

A261780 Number A(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 4, 0, 1, 4, 15, 24, 8, 0, 1, 5, 26, 73, 82, 16, 0, 1, 6, 40, 164, 354, 280, 32, 0, 1, 7, 57, 310, 1031, 1716, 956, 64, 0, 1, 8, 77, 524, 2395, 6480, 8318, 3264, 128, 0, 1, 9, 100, 819, 4803, 18501, 40728, 40320, 11144, 256, 0

Offset: 0

Alois P. Heinz, Aug 31 2015

Also the number of k-compositions of n: matrices with k rows of nonnegative integers with positive column sums and total element sum n.
A(2,2) = 7: (matrices and corresponding marked compositions are given)
  [1 1]  [0 0]  [1 0]  [0 1]  [1]   [2]   [0]
  [0 0]  [1 1]  [0 1]  [1 0]  [1]   [0]   [2]
  1a1a,  1b1b,  1a1b,  1b1a,  2ab,  2aa,  2bb.

			A(3,2) = 24: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1, ...
  0,  1,   2,    3,     4,      5,      6, ...
  0,  2,   7,   15,    26,     40,     57, ...
  0,  4,  24,   73,   164,    310,    524, ...
  0,  8,  82,  354,  1031,   2395,   4803, ...
  0, 16, 280, 1716,  6480,  18501,  44022, ...
  0, 32, 956, 8318, 40728, 142920, 403495, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
E. Grazzini, E. Munarini, M. Poneti, S. Rinaldi, m-compositions and m-partitions: exhaustive generation and Gray code, Pure Math. Appl. 17 (2006), 111-121.
G. Louchard, Matrix Compositions: a Probabilistic analysis, Proc. GASCOM'08, Pure Mathematics and Applications, 19, 2-3, 127-146, 2008.
E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.8

Columns k=0-10 give: A000007, A011782, A003480, A145839, A145840, A145841, A161434, A261799, A261800, A261801, A261802.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A261783.
Cf. A261718 (same for partitions), A261781.

A:= proc(n, k) option remember; `if`(n=0, 1,
      add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[n_, k_] := SeriesCoefficient[(1-x)^k/(2*(1-x)^k-1), {x, 0, n}]; Table[ a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 07 2017 *)

G.f. of column k: (1-x)^k/(2*(1-x)^k-1).
A(n,k) = Sum_{i=0..k} C(k,i) * A261781(n,k-i).
A(n,k) = Sum_{j>=0} (1/2)^(j+1) * binomial(n-1+k*j,n). - Seiichi Manyama, Aug 06 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

A374455 Expansion of o.g.f. 1/(1 - 5x - 10x^2 - 10x^3 - 5x^4 - x^5).

Original entry on oeis.org

1, 5, 35, 235, 1580, 10626, 71460, 480570, 3231845, 21734235, 146163251, 982951365, 6610371480, 44454906580, 298960311840, 2010515259876, 13520763292345, 90927457083265, 611489327404315, 4112280377388895, 27655184063541876, 185981775414350150, 1250731895575163300

Offset: 0

Enrique Navarrete, Jul 08 2024

a(n) is the number of generalized compositions of n using parts of size at most 5 where there are binomial(5,i) types of i (see example).

The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i.

			The following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 5 using parts of size at most 5 where there are binomial(5,i) types of i (ie. 5 types of 1, 10 types of 2, 10 types of 3, 5 types of 4, and 1 type of 5):
    Type                     Number              Total
    5                          1                     1
    4+1                        2                    50
    3+2                        2                   200
    3+1+1                      3                   750
    2+2+1                      3                  1500
    2+1+1+1                    4                  5000
    1+1+1+1+1+1                1                  3125,
    adding to a(5)=10626.

Index entries for linear recurrences with constant coefficients, signature (5,10,10,5,1).

Cf. A000012, A000129, A108368, A374454.

Cf. A145841.

CoefficientList[Series[1/(1-5*x-10*x^2-10*x^3-5*x^4-x^5),{x,0,22}],x] (* Stefano Spezia, Jul 09 2024 *)

a(n) = 5*a(n-1) + 10*a(n-2) + 10*a(n-3) + 5*a(n-4) + a(n-5), n=>5.

a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(5*k,n). - Seiichi Manyama, Aug 03 2024

a(20) corrected by Georg Fischer, Oct 28 2024

cp's OEIS Frontend

A003480 a(0) = 1, a(1) = 2, for n > 1, a(n) = 4*a(n-1) - 2*a(n-2).

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A261780 Number A(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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A374455 Expansion of o.g.f. 1/(1 - 5*x - 10*x^2 - 10*x^3 - 5*x^4 - x^5).

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A003480 a(0) = 1, a(1) = 2, for n > 1, a(n) = 4a(n-1) - 2a(n-2).

A374455 Expansion of o.g.f. 1/(1 - 5x - 10x^2 - 10x^3 - 5x^4 - x^5).