1,5

Row sums are the cake numbers, A000125. Alternating sum of row n is 0 if n even and (3-n)/2 if n odd. Rows are symmetric, beginning and ending with 1. The number of occurrences of k in this triangle is the number of divisors of k-1, given by A000005.

The triangle can be generated by numbers of the form k*(n-k) + 1 for k = 0 to n. Conjecture: except for n = 0,1 and 6 every row contains a prime. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 15 2005

Comments from Moshe Newman (mshnoiman(AT)hotmail.com), Apr 06 2008: (Start) Consider the semigroup of words in x,y,q subject to the relationships: yx = xyq, qx = xq, qy = yq

Now take words of length n in x and y, with exactly k y's. If there had been no relationships, the number of different words of this type would be n choose k, sequence A007318. Thanks to the relationships, the number of words of this type is the k-th entry in the n-th row of this sequence (read as a triangle, with the first row indexed by zero and likewise the first entry in each row.)

For example: with three letters and one y, we have three possibilities: xxy, xyx = xxyq, yxx = xxyqq. No two of them are equal, so this entry is still 3, as in Pascal's triangle.

With four letters, two y's, we have the first reduction: xyyx = yxxy = xxyyqq and this is the only reduction for 4 letters. So the middle entry of the fourth row is 5 instead of 6, as in the Pascal triangle. (End)

t(i, j)=(i-j)(j-1)+1.

As a square array read by antidiagonals, a(n, k) = 1+n*k. a(n, k)=a(n-1, k)+k. Row n has g.f. (1+(n-1)x)/(1-x)^2, n>=0. - Paul Barry (pbarry(AT)wit.ie), Feb 22 2003

Third diagonal (1,3,5,7,...) consists of the positive integers congruent to 1 mod 2.

Triangle begins:

1

1 1

1 2 1

1 3 3 1

1 4 5 4 1

1 5 7 7 5 1

1 6 9 10 9 6 1

...

As a square array read by antidiagonals, the first rows are:

1 1 1. 1. 1. 1 ...

1 2 3. 4. 5. 6 ...

1 3 5. 7. 9 11 ...

1 4 7 10 13 16 ...

1 5 9 13 17 21 ...

Cf. A077029, A003991.

The maximum value for each anti-diagonal is sequence A033638.

A004247(n) + 1.

Sequence in context: A099573 A107430 A132892 this_sequence A114225 A072704 A038792

Adjacent sequences: A077025 A077026 A077027 this_sequence A077029 A077030 A077031

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Oct 19 2002