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A111373 A generalized Pascal triangle. 2
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, 0, 143, 0, 0, 130, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

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, Nov 10 2005

Row sums are A126042. - Paul Barry, Dec 16 2006

LINKS

Table of n, a(n) for n=0..97.

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

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

FORMULA

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

From Paul Barry, Dec 16 2006: (Start)

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. (End)

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

G.f. (x*A(x))^k=sum{n>=k, T(n,k)*x^n}, where A(x)=1+x^3*A(x)^3. - Vladimir Kruchinin, Feb 18 2011

EXAMPLE

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

Production matrix is

0, 1,

0, 0, 1,

1, 0, 0, 1,

0, 1, 0, 0, 1,

0, 0, 1, 0, 0, 1,

0, 0, 0, 1, 0, 0, 1,

0, 0, 0, 0, 1, 0, 0, 1,

0, 0, 0, 0, 0, 1, 0, 0, 1,

0, 0, 0, 0, 0, 0, 1, 0, 0, 1

CROSSREFS

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

Cf. A000012, A000027, A027379, A111396.

Sequence in context: A212193 A253189 A126030 * A336757 A116376 A165766

Adjacent sequences:  A111370 A111371 A111372 * A111374 A111375 A111376

KEYWORD

nonn,easy,tabl

AUTHOR

N. J. A. Sloane, Nov 09 2005

EXTENSIONS

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

STATUS

approved

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Last modified May 18 23:41 EDT 2021. Contains 344009 sequences. (Running on oeis4.)