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A204057 Triangle derived from an array of f(x), Narayana polynomials. 2
1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 14, 1, 1, 5, 19, 45, 42, 1, 1, 6, 29, 100, 197, 132, 1, 1, 7, 41, 185, 562, 903, 429, 1, 1, 8, 55, 306, 1257, 3304, 4279, 1430, 1, 1, 9, 71, 469, 2426, 8925, 20071, 20793, 4862, 1, 1, 10, 89, 680, 4236, 20076, 65445 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums = (1, 2, 4, 10, 31, 113, 466, 2129, 10641, 138628, 335379, 2702364,...)

Another version of triangle in A008550. - Philippe Deléham, Jan 13 2012

Another version of A243631. - Philippe Deléham, Sep 26 2014

LINKS

Table of n, a(n) for n=1..62.

FORMULA

The triangle is the set of antidiagonals of an array in which columns are f(x) of the Narayana polynomials; with column 1 = (1, 1, 1,...) column 2 = (1, 2, 3,..), column 3 = A028387, column 4 = A090197, then A090198, A090199,...

The array by rows is generated from production matrices of the form:

1, (N-1)

1, 1, (N-1)

1, 1, 1, (N-1)

1, 1, 1, 1, (N-1)

...(infinite square matrices with the rest zeros); such that if the matrix is M, n-th term in row N is the upper left term of M^n.

EXAMPLE

First few rows of the array =

1,....1,....1,.....1,.....1,...;

1.....2,....5,....14,....42,...; = A000108

1,....3,...11,....45,...197,...; = A001003

1,....4,...19,...100,...562,...; = A007564

1,....5,...29,...185,..1257,...; = A059231

1,....6,...41,...306,..2426,...; = A078009

...

First few rows of the triangle =

1;

1, 1;

1, 2, 1;

1, 3, 5, 1;

1, 4, 11, 14, 1;

1, 5, 19, 45, 42, 1;

1, 6, 29, 100, 197, 132, 1;

1, 7, 41, 185, 562, 903, 429, 1;

1, 8, 55, 306, 1257, 3304, 4279, 1430, 1;

1, 9, 71, 469, 2426, 8952, 20071, 20793, 4862, 1;

...

Examples: column 4 of the array = A090197: (1, 14, 45, 100,...) = N(4,n) where N(4,x) is the 4th Narayana polynomial.

Term (5,3) = 29 is the upper left term of M^3, where M = the infinite square production matrix:

1, 4, 0, 0, 0,...

1, 1, 4, 0, 0,...

1, 1, 1, 4, 0,...

1, 1, 1, 1, 4,...

... generating row 5, A059231: (1, 5, 29, 185,...).

CROSSREFS

Cf. A000108, A028387, A090197, A090198, A090199, A090200, A001003, A007564, A059231, A078009.

Cf. A008550, A243631.

Sequence in context: A143327 A094954 A083064 * A335975 A241578 A112338

Adjacent sequences:  A204054 A204055 A204056 * A204058 A204059 A204060

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jan 09 2012

EXTENSIONS

Corrected by Philippe Deléham, Jan 13 2012

STATUS

approved

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Last modified September 25 10:30 EDT 2020. Contains 337337 sequences. (Running on oeis4.)