login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A082907
A modified Pascal's triangle, read by rows, and modified as follows: binomial(n,j) is replaced by gcd(2^n, binomial(n,j)), i.e., the largest power of 2 dividing binomial(n,j).
10
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 2, 4, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 8, 2, 8, 4, 8, 1, 1, 1, 4, 4, 2, 2, 4, 4, 1, 1, 1, 2, 1, 8, 2, 4, 2, 8, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 4, 2, 4, 1, 8, 4, 8, 1, 4, 2, 4, 1, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1
OFFSET
0,5
COMMENTS
If N is a power of 2, then the first N rows are invariant under all 6 symmetries of an equilateral triangle. - Paul Boddington, Dec 17 2003
LINKS
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
E. Burlachenko, Fractal generalized Pascal matrices, arXiv:1612.00970 [math.NT], 2016. See p. 5.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
FORMULA
From Paul Boddington, Dec 17 2003: (Start)
T(n, j) = c(n)/(c(j)*c(n-j)) where c(n)=A060818(n).
T(n, j) = (b(j)*b(n-j))/b(n) where b(n)=A001316(n) (Gould's sequence). (End)
EXAMPLE
Triangle read by rows:
1,
1,1,
1,2,1,
1,1,1,1,
1,4,2,4,1,
1,1,2,2,1,1,
1,2,1,4,1,2,1,
1,1,1,1,1,1,1,1,
1,8,4,8,2,8,4,8,1,
1,1,4,4,2,2,4,4,1,1,
...
For n = -1 + 2^k, such rows consist of all 1's since all binomial coefficients C(n,j) are odd.
MATHEMATICA
Flatten[Table[Table[GCD[2^n, Binomial[n, j]], {j, 0, n}], {n, 0, 25}], 1]
f[n_] := Denominator[CatalanNumber[n - 1]/2^(n - 1)]; T[n_, k_] := f[n]/(f[k]*f[n - k]); Table[T[n, k], {n, 0, 7}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 24 2016 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Labos Elemer, Apr 23 2003
EXTENSIONS
Edited by Jon E. Schoenfield, Dec 24 2016
STATUS
approved