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A145839 Number of 3-compositions of n.

%I A145839 #34 May 30 2025 10:32:13
%S A145839 1,3,15,73,354,1716,8318,40320,195444,947380,4592256,22260144,
%T A145839 107902088,523036176,2535324816,12289536016,59571339552,288761470848,
%U A145839 1399719859808,6784893012864,32888561860032,159421452802624,772767131681280,3745851196992000
%N A145839 Number of 3-compositions of n.
%C A145839 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.
%C A145839 Matrix inverse of (A000217(A004736)*A154990). - _Mats Granvik_, Jan 19 2009
%C A145839 (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
%C A145839 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
%D A145839 G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.
%H A145839 Alois P. Heinz, <a href="/A145839/b145839.txt">Table of n, a(n) for n = 0..1000</a>
%H A145839 Sela Fried, <a href="https://arxiv.org/abs/2505.14196">Even-up words and their variants</a>, arXiv:2505.14196 [math.CO], 2025. See p. 8.
%H A145839 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H A145839 Emanuele Munarini, Maddalena Poneti, and Simone Rinaldi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Rinaldi/rinaldi.html">Matrix compositions</a>, JIS 12 (2009) 09.4.8.
%H A145839 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6,2).
%F A145839 a(n+3) = 6*a(n+2) - 6*a(n+1) + 2*a(n).
%F A145839 G.f.: (1-x)^3/(2*(1-x)^3 - 1).
%F A145839 a(n) = Sum_{k>=0} C(n+3*k-1,n) / 2^(k+1). - _Vaclav Kotesovec_, Dec 31 2013
%F A145839 a(n) = Sum_{j=0..n-1} binomial(n-j+2, 2)*a(j) with a(0) = 1. - _G. C. Greubel_, Mar 07 2021
%p A145839 a:= proc(n) option remember; `if`(n=0, 1,
%p A145839       add(a(n-j)*binomial(j+2, 2), j=1..n))
%p A145839     end:
%p A145839 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 01 2015
%t A145839 Table[Sum[Binomial[n+3*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* _Vaclav Kotesovec_, Dec 31 2013 *)
%t A145839 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 *)
%o A145839 (Sage)
%o A145839 @CachedFunction
%o A145839 def a(n):
%o A145839     if n==0: return 1
%o A145839     else: return sum( binomial(n-j+2,2)*a(j) for j in (0..n-1))
%o A145839 [a(n) for n in (0..25)] # _G. C. Greubel_, Mar 07 2021
%o A145839 (Magma) 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
%Y A145839 Cf. A003480 (2-compositions), A145840 (4-compositions), A145841 (5-compositions).
%Y A145839 Column k=3 of A261780.
%K A145839 nonn,easy
%O A145839 0,2
%A A145839 Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008
%E A145839 Offset corrected by _Alois P. Heinz_, Aug 31 2015