

A327889


Triangle read by rows T(n, k) = (1)^k * A000217(k) * F(binomial(n,k)), where F(x) = 1 if the largest decimal digit of x is 1, and 0 otherwise.


2



0, 0, 1, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0, 0, 10, 0, 0, 3, 6, 0, 15, 0, 0, 3, 0, 10, 0, 21, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 10, 15, 0, 0, 0, 45, 0, 1, 0, 6, 0, 0, 0, 28, 0, 45, 55, 0, 1, 0, 6, 0, 0, 0, 0, 36, 0, 55, 66, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 66, 78
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OFFSET

1,6


COMMENTS

Represents alternating, normalized (linear) modification of A327853, transformed by first (decimal) digit indicator function F(x).
The scatterplot of the sequence represents a fractallike structure, made out of arclike structures. For comparison, the A327853 represents a Sierpinski's gasket (triangle), bounded by a function of A003056 (the positive inverse of triangular numbers).
If a number base other than decimal is used, then for a larger value of the base, the scatterplot structure will appear to be "zoomed in". The smallest base that will still represent the structure is ternary, since in binary we have F(x)=1 for all x, and the scatterplot will degrade to a simple triangle structure.
If we modify F(x) to look at other digits than the largest digit, then the structure appears to lose "density".
Why does Pascal's triangle (Sierpinski's gasket) converge to such arclike structure when the digit indicator function F(x) is applied (in some number base)? Are there sequences other than those related to binomial coefficients, that can replicate this structure?


LINKS

Matej Veselovac, Table of n, a(n) for n = 1..100000
Math StackExchange, Pattern in Pascal's triangle .
Matej Veselovac, Scatterplot of the sequence, for terms a(n), n=1...10^5..


FORMULA

The entries of the triangle are given by T(n, k) = (1)^k * A000217(k) * F(binomial(n,k)), then it is read by rows, where F(x) = 1 if the largest decimal digit of x is 1, and 0 otherwise.


EXAMPLE

First 16 rows of the T(n, k):
0;
0, 1;
0, 0, 3;
0, 0, 0, 6;
0, 0, 0, 0, 10;
0, 0, 3, 6, 0, 15;
0, 0, 3, 0, 10, 0, 21;
0, 0, 0, 0, 0, 0, 0, 28;
0, 0, 0, 0, 0, 0, 0, 0, 36;
0, 0, 0, 0, 10, 15, 0, 0, 0,45;
0, 1, 0, 6, 0, 0, 0, 28, 0,45, 55;
0, 1, 0, 6, 0, 0, 0, 0, 36, 0, 55, 66;
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 66, 78;
0, 1, 0, 0, 0, 15, 21, 28, 36, 0, 0, 0, 78, 91;
0, 1, 0, 0, 10, 0, 0, 0, 0, 0, 55, 0, 0, 91, 105;
0, 1, 3, 0, 10, 0, 0, 0, 0, 0, 0, 66, 0, 91, 105, 120;


MATHEMATICA

d[n_, b_: 10] := Mod[Floor[n/b^(Floor[Log[b, n]])], b]; t[n_] := n (n + 1)/2; f[x_] := x ; r[n0_, b_: 10] := Flatten[Table[(1)^k Floor[f[t[k]]]*If[d[Binomial[n, k], b] == 1, 1, 0], {n, 0, n0}, {k, 0, n}]]; r[20] (* Matej Veselovac, Sep 29 2019 *)


CROSSREFS

Cf. A001317, A007318, A003056, A000217.
Cf. A327853 (original sequence, before applying the transformation).
Sequence in context: A081805 A333784 A181189 * A221702 A084681 A261038
Adjacent sequences: A327886 A327887 A327888 * A327890 A327891 A327892


KEYWORD

sign,base,tabl,look


AUTHOR

Matej Veselovac, Sep 29 2019


STATUS

approved



