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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).

%I A082907 #26 Dec 25 2016 02:13:51
%S A082907 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,
%T A082907 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,
%U A082907 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
%N 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).
%C A082907 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
%H A082907 G. C. Greubel, <a href="/A082907/b082907.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A082907 Tyler Ball, Tom Edgar, and Daniel Juda, <a href="http://dx.doi.org/10.4169/math.mag.87.2.135">Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem</a>, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
%H A082907 E. Burlachenko, <a href="https://arxiv.org/abs/1612.00970">Fractal generalized Pascal matrices</a>, arXiv:1612.00970 [math.NT], 2016. See p. 5.
%H A082907 Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
%F A082907 From _Paul Boddington_, Dec 17 2003: (Start)
%F A082907 T(n, j) = c(n)/(c(j)*c(n-j)) where c(n)=A060818(n).
%F A082907 T(n, j) = (b(j)*b(n-j))/b(n) where b(n)=A001316(n) (Gould's sequence). (End)
%e A082907 Triangle read by rows:
%e A082907             1,
%e A082907            1,1,
%e A082907           1,2,1,
%e A082907          1,1,1,1,
%e A082907         1,4,2,4,1,
%e A082907        1,1,2,2,1,1,
%e A082907       1,2,1,4,1,2,1,
%e A082907      1,1,1,1,1,1,1,1,
%e A082907     1,8,4,8,2,8,4,8,1,
%e A082907    1,1,4,4,2,2,4,4,1,1,
%e A082907   ...
%e A082907 For n = -1 + 2^k, such rows consist of all 1's since all binomial coefficients C(n,j) are odd.
%t A082907 Flatten[Table[Table[GCD[2^n, Binomial[n, j]], {j, 0, n}], {n, 0, 25}], 1]
%t A082907 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 *)
%Y A082907 Cf. A000005, A000079, A001316, A007318, A060818.
%K A082907 nonn,tabl
%O A082907 0,5
%A A082907 _Labos Elemer_, Apr 23 2003
%E A082907 Edited by _Jon E. Schoenfield_, Dec 24 2016