A243756 Triangle read by rows: T(n,k) = A242954(n)/(A242954(k) * A242954(n-k)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 4, 4, 4, 1, 1, 1, 4, 4, 1, 1, 4, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 4, 4, 4, 1, 4, 4, 4, 1

Offset

0,12

Comments

The exponent of T(n,k) is the number of 'carries' that occur when adding k and n-k in base 4 using the traditional addition algorithm.

If T(n,k) != 0 mod 4, then n dominates k in base 4.

Links

Table of n, a(n) for n=0..90.

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.

Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

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

T(n,k) = A242954(n)/(A242954(k) * A242954(n-k)).

T(n,k) = Product_{i=1..n} A234957(i)/(Product_{i=1..k} A234957(i)*Product_{i=1..n-k} A234957(i)).

T(n,k) = A234957(n)/n*(k/A234957(k)*T(n-1,k-1)+(n-k)/A234957(n-k)*T(n-1,k)).

Example

The triangle begins:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 4, 4, 4, 1;
1, 1, 4, 4, 1, 1;
1, 1, 1, 4, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
1, 4, 4, 4, 1, 4, 4, 4, 1;
1, 1, 4, 4, 1, 1, 4, 4, 1, 1;
1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Prog

(Sage)
m=50
T=[0]+[4^valuation(i, 4) for i in [1..m]]
Table=[[prod(T[1:i+1])/(prod(T[1:j+1])*prod(T[1:i-j+1])) for j in [0..i]] for i in [0..m-1]]
[x for sublist in Table for x in sublist]

Crossrefs

Cf. A082907, A234957, A242849, A242954.

Sequence in context: A176101 A052283 A133889 * A172985 A117683 A172088

Adjacent sequences: A243753 A243754 A243755 * A243757 A243758 A243759

Keyword

nonn,tabl

Author

Tom Edgar, Jun 09 2014

Status

approved