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Symmetrical triangle of coefficients based on invert transform of A001906.
1

%I #40 Sep 08 2022 08:45:35

%S 1,3,3,8,9,8,21,24,24,21,55,63,64,63,55,144,165,168,168,165,144,377,

%T 432,440,441,440,432,377,987,1131,1152,1155,1155,1152,1131,987,2584,

%U 2961,3016,3024,3025,3024,3016,2961,2584,6765,7752,7896,7917,7920,7920,7917,7896,7752,6765

%N Symmetrical triangle of coefficients based on invert transform of A001906.

%C Row sums are {1, 6, 25, 90, 300, 954, 2939, 8850, 26195, 76500, ...}.

%C It can be noticed that the interior of the triangle is relatively "flat", which is a smaller variation than in most symmetrical triangles of this type.

%C 16*T(n,k) is the number of Boolean (equivalently, lattice, modular lattice, distributive lattice) intervals of the form [s_{k+1},w] in the Bruhat order on S_{n+3}, for the simple reflection s_{k+1}. - _Bridget Tenner_, Jan 16 2020

%H G. C. Greubel, <a href="/A141678/b141678.txt">Rows n=1..101 of triangle, flattened</a>

%H Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, <a href="https://arxiv.org/abs/2203.13205">Honeycombs in the Pascal triangle and beyond</a>, arXiv:2203.13205 [math.HO], 2022. See p. 5.

%H B. E. Tenner, <a href="https://arxiv.org/abs/2001.05011">Interval structures in the Bruhat and weak orders</a>, arXiv:2001.05011 [math.CO], 2020.

%F Let b(n) = Sum_{k=1..n} k*b(n - k), then T(n, m) = b(n-m+1)*b(m+1).

%F Alternatively, let f(n) = Fibonacci(2*n) with f(0)=1, then T(n, k) = f(n-k+1)*f(k+1). - _G. C. Greubel_, Apr 06 2019

%e Triangle begins as:

%e 1;

%e 3, 3;

%e 8, 9, 8;

%e 21, 24, 24, 21;

%e 55, 63, 64, 63, 55;

%e 144, 165, 168, 168, 165, 144;

%e 377, 432, 440, 441, 440, 432, 377; ...

%t b[0]=1; b[n_]:= Sum[k*b[n-k], {k, 1, n}];

%t Table[b[n-m+1]*b[m+1], {n, 0, 10}, {m, 0, n}]//Flatten

%t f[n_]:= If[n == 0, 1, Fibonacci[2*n]]; Table[f[n-k+1]*f[k+1], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 06 2019 *)

%o (PARI) {b(n) = if(n==0, 1, fibonacci(2*n))};

%o for(n=0, 10, for(k=0, n, print1(b(n-k+1)*b(k+1), ", "))) \\ _G. C. Greubel_, Apr 06 2019

%o (Magma) b:= func< n| n eq 0 select 1 else Fibonacci(2*n) >; [[b(n-k+1)*b(k+1): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Apr 06 2019

%o (Sage)

%o @CachedFunction

%o def b(n):

%o if n==0: return 1

%o return fibonacci(2*n)

%o [[b(n-k+1)*b(k+1) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 06 2019

%Y Cf. A001906.

%K nonn

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 07 2008

%E Edited by _G. C. Greubel_, Apr 02 2019