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A145839 Number of 3-compositions of n. 5
1, 3, 15, 73, 354, 1716, 8318, 40320, 195444, 947380, 4592256, 22260144, 107902088, 523036176, 2535324816, 12289536016, 59571339552, 288761470848, 1399719859808, 6784893012864, 32888561860032, 159421452802624, 772767131681280, 3745851196992000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 + 3x + 15x^2 + 73x^3 + ...) = 1/(1-3x-6x^2-10x^3-15x^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]

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

M. Janjic, 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, S. Rmialdi, Matrix compositions, JIS 12 (2009) 09.4.8

FORMULA

a(n+3) = 6a(n+2)-6a(n+1)+2a(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

MAPLE

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

MATHEMATICA

Table[Sum[Binomial[n+3*k-1, n]/2^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 31 2013 *)

CROSSREFS

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

Column k=3 of A261780.

Sequence in context: A155117 A137638 A156019 * A232289 A055837 A124543

Adjacent sequences:  A145836 A145837 A145838 * A145840 A145841 A145842

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Offset corrected by Alois P. Heinz, Aug 31 2015

STATUS

approved

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Last modified April 18 16:32 EDT 2019. Contains 322209 sequences. (Running on oeis4.)