A261780 -id:A261780 - 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

A261781 Number T(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 and all k letters occur at least once in the composition; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 4, 16, 13, 0, 8, 66, 132, 75, 0, 16, 248, 924, 1232, 541, 0, 32, 892, 5546, 13064, 13060, 4683, 0, 64, 3136, 30720, 114032, 195020, 155928, 47293, 0, 128, 10888, 162396, 893490, 2327960, 3116220, 2075948, 545835

Offset: 0

Alois P. Heinz, Aug 31 2015

From Vaclav Kotesovec, Oct 14 2017: (Start)
Conjecture: For k > 0 the recurrence order for column k is equal to k*(k+1)/2.
Column k > 0 is asymptotic to c(k) * d(k)^n, where c(k) and d(k) are constants (dependent only on k).
k                           c(k)                            d(k)
1  A131577(n) ~ 0.50000000000000000000000000 * 2.00000000000000000000000000^n.
2  A293579(n) ~ 0.60355339059327376220042218 * 3.41421356237309504880168872^n.
3  A293580(n) ~ 0.64122035031051210658648604 * 4.84732210186307263951891624^n.
4  A293581(n) ~ 0.66065168848540565019767995 * 6.28521350788324520158143964^n.
5  A293582(n) ~ 0.67250239588725756267924287 * 7.72502395887257562679242875^n.
6  A293583(n) ~ 0.68048292906885160660288253 * 9.16579514882621927923459043^n.
7  A293584(n) ~ 0.68622254929933439577377124 * 10.6071156901906815408327973^n.
8  A293585(n) ~ 0.69054873168854973836384871 * 12.0487797070167958138215794^n.
9  A293586(n) ~ 0.69392626461456654033893782 * 13.4906727630621977261008808^n.
10 A293587(n) ~ 0.69663630864564830007443110 * 14.9327261729129660014886221^n.
---
Conjecture: d(k+1) - d(k) tends to 1/log(2).
d(2) - d(1) = 1.414213562373095048801688724209698...
d(3) - d(2) = 1.433108539489977590717227522340838...
d(4) - d(3) = 1.437891406020172562062523400686067...
d(5) - d(4) = 1.439810450989330425210989107036901...
d(6) - d(5) = 1.440771189953643652442161677346934...
d(7) - d(6) = 1.441320541364462261598206961226199...
d(8) - d(7) = 1.441664016826114272988782079622148...
d(9) - d(8) = 1.441893056045401912279301345910755...
d(10)- d(9) = 1.442053409850768275387741352145193...
1 / log(2)  = 1.442695040888963407359924681001892...
(End)

			A(3,2) = 16: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,    3;
  0,  4,   16,    13;
  0,  8,   66,   132,     75;
  0, 16,  248,   924,   1232,    541;
  0, 32,  892,  5546,  13064,  13060,   4683;
  0, 64, 3136, 30720, 114032, 195020, 155928, 47293;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
E. Munarini, M. Poneti, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Table 2.

Columns k=0..10 give A000007, A131577, A293579, A293580, A293581, A293582, A293583, A293584, A293585, A293586, A293587.
Row sums give A120733.
Main diagonal gives A000670.
T(2n,n) gives A261784.
T(n+1,n)/2 gives A083385.
Cf. A261719 (same for partitions), A261780.

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:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n==0, 1,
    Sum[A[n-j, k]*Binomial[j+k-1, k-1], {j, 1, n}]];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261780(n,k-i).

A261718 Number A(n,k) of partitions 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, 3, 0, 1, 4, 15, 18, 5, 0, 1, 5, 26, 55, 50, 7, 0, 1, 6, 40, 124, 216, 118, 11, 0, 1, 7, 57, 235, 631, 729, 301, 15, 0, 1, 8, 77, 398, 1470, 2780, 2621, 684, 22, 0, 1, 9, 100, 623, 2955, 8001, 12954, 8535, 1621, 30, 0

Offset: 0

Alois P. Heinz, Aug 29 2015

			A(3,2) = 18: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1,       1, ...
  0,  1,   2,    3,     4,      5,      6,       7, ...
  0,  2,   7,   15,    26,     40,     57,      77, ...
  0,  3,  18,   55,   124,    235,    398,     623, ...
  0,  5,  50,  216,   631,   1470,   2955,    5355, ...
  0,  7, 118,  729,  2780,   8001,  19158,   40299, ...
  0, 11, 301, 2621, 12954,  45865, 130453,  317905, ...
  0, 15, 684, 8535, 55196, 241870, 820554, 2323483, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000041, A074141, A261737, A261738, A261739, A261740, A261741, A261742, A261743, A261744.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A209668.
Cf. A144064, A261719, A261780.

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

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k]*Binomial[i + k - 1, k - 1]]]]; A[n_, k_] := b[n, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..k} C(k,i) * A261719(n,k-i).

A261835 Number A(n,k) of compositions of n into distinct parts 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, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 16, 3, 0, 1, 5, 10, 46, 21, 5, 0, 1, 6, 15, 100, 75, 50, 11, 0, 1, 7, 21, 185, 195, 231, 205, 13, 0, 1, 8, 28, 308, 420, 736, 1414, 292, 19, 0, 1, 9, 36, 476, 798, 1876, 6032, 2376, 587, 27, 0

Offset: 0

Alois P. Heinz, Sep 02 2015

Also matrices with k rows of nonnegative integers with distinct positive column sums and total element sum n.
A(2,2) = 3: (matrices and corresponding marked compositions are given)
  [1]   [2]   [0]
  [1]   [0]   [2]
  2ab,  2aa,  2bb.

			A(3,2) = 16: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,      1,      1, ...
  0,  1,   2,    3,     4,     5,      6,      7, ...
  0,  1,   3,    6,    10,    15,     21,     28, ...
  0,  3,  16,   46,   100,   185,    308,    476, ...
  0,  3,  21,   75,   195,   420,    798,   1386, ...
  0,  5,  50,  231,   736,  1876,   4116,   8106, ...
  0, 11, 205, 1414,  6032, 19320,  51114, 117936, ...
  0, 13, 292, 2376, 11712, 42610, 126288, 322764, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A032020, A261840, A261841, A261842, A261843, A261844, A261845, A261846, A261847, A261848.
Rows n=0-4 give: A000012, A001477, A000217, A255211, A228317(n+2).
Main diagonal gives A261837.
Cf. A261780, A261836.

b:= proc(n, i, p, k) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
    end:
A:= (n, k)-> b(n$2, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2 < n, 0, If[n == 0, p!, b[n, i-1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; A[n_, k_] := b[n, n, 0, k]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 16 2017, translated from Maple *)

A(n,k) = Sum_{i=0..k} C(k,i) * A261836(n,k-i).

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

A261783 Number of compositions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order.

Original entry on oeis.org

1, 1, 7, 73, 1031, 18501, 403495, 10366833, 306717703, 10271072557, 384058268507, 15861842372465, 717135437119271, 35228475333207937, 1868440035684996207, 106412817671933423073, 6477200889282232394759, 419626639092214594301373, 28829330550533269570699411

Offset: 0

Alois P. Heinz, Aug 31 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..365

Main diagonal of A261780.

Cf. A209668.

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:
a:= n-> A(n$2):
seq(a(n), n=0..20);

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n - j, k]*Binomial[j + k - 1, k - 1], {j, 1, n}]]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2017, translated from Maple *)

a(n) = A261780(n,n).
a(n) = [x^n] (1-x)^n/(2*(1-x)^n-1).
a(n) ~ n^n / (sqrt(2) * (log(2))^(n+1)). - Vaclav Kotesovec, Sep 21 2019
a(n) = Sum_{k>=1} (1/2)^k * binomial(k*n-1,n). - Seiichi Manyama, Aug 06 2024

A382923 Square array A(n,k), n >= 0, k >= 0, read by downward antidiagonals: A(n,k) is the number of m-compositions of n with k zeros.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 5, 7, 0, 4, 13, 16, 16, 0, 5, 14, 33, 40, 35, 0, 6, 29, 70, 105, 100, 75, 0, 7, 27, 88, 207, 292, 244, 159, 0, 8, 51, 152, 336, 604, 758, 576, 334, 0, 9, 44, 206, 588, 1161, 1749, 1920, 1329, 696, 0, 10, 79, 300, 882, 2076, 3685, 4924, 4802, 3028, 1442

Offset: 0

John Tyler Rascoe, Apr 09 2025

For some m > 0, an m-composition of n is a rectangular array of nonnegative integers with m rows, at least one nonzero entry in each column, and having the sum of all entries equal to n.

			Square array begins:
   1,   0,   0,   0,    0,    0, ...
   1,   2,   3,   4,    5,    6, ...
   3,   5,  13,  14,   29,   27, ...
   7,  16,  33,  70,   88,  152, ...
  16,  40, 105, 207,  336,  588, ...
  35, 100, 292, 604, 1161, 2076, ...
  ...
A(2,0) = 3 counts:
  [2],  [1,1],  [1]
                [1].
A(2,1) = 5 counts:
  [2]   [0]   [1]   [1]   [0]
  [0],  [2],  [1]   [0]   [1]
              [0],  [1],  [1].

John Tyler Rascoe, Antidiagonals n = 0..30, flattened

Cf. A038207, A101509 (column k=0), A181331, A261780, A323429, A382924 (main diagonal).

G_tx(max_row) = {my(row = max_row, N = row*2, m = List([concat([1],vector(row-1,i,0))]), x='x+O('x^N), h=1 + sum(m=1,N,-1+ 1/(1 + t^m - (t + x/(1-x))^m))); for(n=1,row, listput(m,Vecrev(polcoeff(h, n))[1..row])); matrix(row, row, i,j, m[i][j])}
G_tx(10)

G.f.: G(t,x) = 1 + Sum_{m>0} -1 + 1/(1 + t^m - (t + x/(1 - x))^m).

A161434 Number of 6-compositions.

Original entry on oeis.org

1, 6, 57, 524, 4803, 44022, 403495, 3698352, 33898338, 310705224, 2847860436, 26102905368, 239253883390, 2192952083712, 20100149570496, 184233853423936, 1688649759962676, 15477817777932456, 141866507103389516, 1300319342589168000, 11918460722228694720

Offset: 0

Emanuele Munarini, Jun 10 2009

Excluding the terms of A161434(0) followed by the INVERTi transform yields A000389 without A000389(0). - Alexander R. Povolotsky and R. J. Mathar, Jun 16 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
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
Index entries for linear recurrences with constant coefficients, signature (12,-30,40,-30,12,-2).

Column k=6 of A261780.

Cf. A000389, A161434.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+5, 5), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 01 2015

Join[{1}, LinearRecurrence[{12, -30, 40, -30, 12, -2}, {6, 57, 524, 4803, 44022, 403495}, 20]] (* Jean-François Alcover, Jan 08 2016 *)
CoefficientList[Series[(1-x)^6/(2*(1-x)^6-1), {x, 0, 50}], x] (* G. C. Greubel, Nov 25 2017 *)

x='x+O('x^30); Vec((1-x)^6/(2*(1-x)^6-1)) \\ G. C. Greubel, Nov 25 2017

Recurrence: a(n+6) = 12*a(n+5) - 30*a(n+4) + 40*a(n+3) - 30*a(n+2) + 12*a(n+1) - 2*a(n).
G.f.: (1-x)^6/(2*(1-x)^6-1).
a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(n-1+6*k,n). - Seiichi Manyama, Aug 06 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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A261781 Number T(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 and all k letters occur at least once in the composition; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A261718 Number A(n,k) of partitions 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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A261835 Number A(n,k) of compositions of n into distinct parts 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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A145841 Number of 5-compositions of n.

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