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A273350 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k up steps (n >= 2, k >= 1). 3

%I

%S 1,1,1,1,3,1,1,6,5,1,1,10,16,7,1,1,15,40,31,9,1,1,21,85,105,51,11,1,1,

%T 28,161,295,219,76,13,1,1,36,280,721,771,396,106,15,1,1,45,456,1582,

%U 2331,1681,650,141,17,1,1,55,705,3186,6244,6083,3235,995,181,19,1

%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k up steps (n >= 2, k >= 1).

%C Number of entries in row n is n-1.

%C Sum of entries in row n = A082582(n).

%C Sum_{k>=1} k*T(n,k) = A273351(n).

%C Connection with A145904 should be explored.

%C This seems to be a duplicate of A145904. - _Alois P. Heinz_, Jun 04 2016. [This could probably be proved by showing that the g.f.s are the same. - _N. J. A. Sloane_, Jul 02 2016]

%H A. Blecher, C. Brennan and A. Knopfmacher, <a href="https://doi.org/10.2989/16073606.2015.1121932">Combinatorial parameters in bargraphs</a>, Quaestiones Mathematicae, 39 (2016), 619-635.

%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.

%F G.f.: G=G(t,z) satisfies zG^2-(1-z-yz-yz^2)G+yz^2=0 (z marks semiperimeter, y marks up-steps).

%F T(n,m) = (1/(n-m))*Sum_{i=0..m} C(n-m,i-1)*C(n-m,i)*C(n-i-1,m-i). - _Vladimir Kruchinin_, Jan 15 2018

%e Row 4 is 1,3,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 1,2,2,2,3 up steps.

%e Triangle starts

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 6, 5, 1;

%e 1, 10, 16, 7, 1;

%e 1, 15, 40, 31, 9, 1.

%p eq := z*G^2-(1-z-y*z-y*z^2)*G+y*z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 18)): for n from 2 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 16 do seq(coeff(P[n], y, j), j = 1 .. n-1) end do; # yields sequence in triangular form

%o (Maxima)

%o T(n,m):=if n<=m then 0 else 1/(n-m)*sum(binomial(n-m,i-1)*binomial(n-m,i)*binomial(n-i-1,m-i),i,0,m); /* _Vladimir Kruchinin_, Jan 15 2018 */

%Y Cf. A082582, A145904, A273351.

%K nonn,tabl

%O 2,5

%A _Emeric Deutsch_, Jun 02 2016

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)