A204057 Triangle derived from an array of f(x), Narayana polynomials.

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

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