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A141678 Symmetrical triangle of coefficients based on invert transform of A001906. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A135477 A182473

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

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Last modified February 1 06:22 EST 2023. Contains 359981 sequences. (Running on oeis4.)