A141678 Symmetrical triangle of coefficients based on invert transform of A001906.

1, 3, 3, 8, 9, 8, 21, 24, 24, 21, 55, 63, 64, 63, 55, 144, 165, 168, 168, 165, 144, 377, 432, 440, 441, 440, 432, 377, 987, 1131, 1152, 1155, 1155, 1152, 1131, 987, 2584, 2961, 3016, 3024, 3025, 3024, 3016, 2961, 2584, 6765, 7752, 7896, 7917, 7920, 7920, 7917, 7896, 7752, 6765

Offset

1,2

Comments

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

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.

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

Links

G. C. Greubel, Rows n=1..101 of triangle, flattened

Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 5.

B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.

Formula

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

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

Example

Triangle begins as:
    1;
    3,   3;
    8,   9,   8;
   21,  24,  24,  21;
   55,  63,  64,  63,  55;
  144, 165, 168, 168, 165, 144;
  377, 432, 440, 441, 440, 432, 377; ...

Mathematica

b[0]=1; b[n_]:= Sum[k*b[n-k], {k, 1, n}];
Table[b[n-m+1]*b[m+1], {n, 0, 10}, {m, 0, n}]//Flatten
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 *)

Prog

(PARI) {b(n) = if(n==0, 1, fibonacci(2*n))};
for(n=0, 10, for(k=0, n, print1(b(n-k+1)*b(k+1), ", "))) \\ G. C. Greubel, Apr 06 2019
(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
(Sage)
@CachedFunction
def b(n):
    if n==0: return 1
    return fibonacci(2*n)
[[b(n-k+1)*b(k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 06 2019

Crossrefs

Cf. A001906.

Sequence in context: A095068 A248696 A021299 * A231855 A368738 A135477

Adjacent sequences: A141675 A141676 A141677 * A141679 A141680 A141681

Keyword

nonn

Author

Roger L. Bagula and Gary W. Adamson, Sep 07 2008

Extensions

Edited by G. C. Greubel, Apr 02 2019

Status

approved