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
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jan 09 2012
EXTENSIONS
Corrected by Philippe Deléham, Jan 13 2012
STATUS
approved