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A204057 Triangle derived from an array of f(x), Narayana polynomials. 5
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, 4237, 20076, 65445, 124996, 103049, 16796, 1 (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

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

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.

From G. C. Greubel, Feb 16 2021: (Start)

T(n, k) = Hypergeometric2F1([1-k, -k], [2], n-k).

Sum_{k=1..n} T(n, k) = A132745(n) - 1. (End)

EXAMPLE

First few rows of the array =

  1,....1,....1,.....1,.....1,...; = A000012

  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,...).

MATHEMATICA

Table[Hypergeometric2F1[1-k, -k, 2, n-k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

PROG

(Sage)

def A204057(n, k): return 1 if n==0 else sum( binomial(n, j)^2*k^j*(n-j)/(n*(j+1)) for j in [0..n-1])

flatten([[A204057(k, n-k) for k in [1..n]] for n in [1..12]]) # G. C. Greubel, Feb 16 2021

(Magma)

A204057:= func< n, k | n eq 0 select 1 else (&+[ Binomial(n, j)^2*k^j*(n-j)/(n*(j+1)): j in [0..n-1]]) >;

[A204057(k, n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 16 2021

CROSSREFS

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

Cf. A008550, A132745, 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 May 24 21:08 EDT 2022. Contains 354043 sequences. (Running on oeis4.)