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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 +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 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, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8. Index entries for linear recurrences with constant coefficients, signature (6,-6,2). FORMULA 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 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 *) 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 *) PROG (Sage) @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 (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 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 June 21 04:47 EDT 2021. Contains 345355 sequences. (Running on oeis4.)