0,12

First diagonal is A000012, the all 1's sequence. Second nonzero diagonal is A000027 = n. Third nonzero diagonal is A027379 = n*(n+5)/2 for n>=1, or essentially A000217(n) - 3. Fourth nonzero diagonal is A111396. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 10 2005

Row sums are A126042. - Paul Barry (pbarry(AT)wit.ie), Dec 16 2006

I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.

Each term is the sum of the two terms above it to the left and two steps to the right.

Riordan array (g(x^3),x*g(x^3)) where g(x)=(2/sqrt(3x))*sin(asin(sqrt(27x/4))/3), the g.f. of A001764; Number triangle T(n,k)=C(3*floor((n+2k)/3)-2k,floor((n+2k)/3)-k)*(k+1)/(2*floor((n+2k)/3)-k+ 1)(2*cos(2*pi*(n-k)/3)+1)/3; - Paul Barry (pbarry(AT)wit.ie), Dec 16 2006

Inverse of Riordan array (1/(1+x^3), x/(1+x^3)), A126030. - Paul Barry (pbarry(AT)wit.ie), Dec 16 2006

Triangle begins:

.1

.0 1

.0 0 1

.1 0 0 1

.0 2 0 0 1

.0 0 3 0 0 1

.3 0 0 4 0 0 1

.0 7 0 0 5 0 0 1

.0 0 12 0 0 6 0 0 1

.12 0 0 18 0 0 7 0 0 1

.0 30 0 0 25 0 0 8 0 0 1

.0 0 55 0 0 33 0 0 9 0 0 1

.55 0 0 88 0 0 42 0 0 10 0 0 1

First column is A001764. Bears same relation to A001764 as A053121 does to A000108.

Cf. A000012, A000027, A027379, A111396.

Sequence in context: A083913 A023670 A126030 this_sequence A116376 A165766 A102082

Adjacent sequences: A111370 A111371 A111372 this_sequence A111374 A111375 A111376

nonn,easy,tabl

N. J. A. Sloane (njas(AT)research.att.com), Nov 09 2005

More terms from Kerri Sullivan (ksulliva(AT)ashland.edu), Jan 23 2006