OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
FORMULA
From Andrew Howroyd, Sep 22 2025: (Start)
T(n,k) = Sum_{j=0..r-1} (r+j)*binomial(k-1, j), where r = n+1-k.
G.f.: x*y*(1 - y*x + y*x^2 - 2*y*x^3)/((1 - x)^2*(1 - y*x - y*x^2)^2). (End)
EXAMPLE
First few rows of the array are:
1, 1, 1, 1, ...
2, 5, 8, 11, ...
3, 7, 16, 30, ...
4, 9, 20, 44, ...
...
First few rows of the triangle are:
1;
2, 1;
3, 5, 1;
4, 7, 8, 1;
5, 9, 16, 11, 1;
6, 11, 20, 30, 14, 1;
7, 13, 24, 44, 49, 17, 1;
...
Example: The diagonal 3, 7, 16, 30, 49, ... (third row of the array) = binomial transform of (3, 4, 5, 0, 0, 0, ...).
PROG
(PARI) T(n, k)=my(r=n+1-k); sum(j=0, r-1, (r+j)*binomial(k-1, j)) \\ Andrew Howroyd, Sep 22 2025
(PARI) T(n) = [Vecrev(p) | p<-Vec((1 - y*x + y*x^2 - 2*y*x^3)/((1 - x)^2*(1 - y*x - y*x^2)^2) + O(x^n))]
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Sep 22 2025
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 16 2006
EXTENSIONS
a(43) corrected and a(46) onwards from Andrew Howroyd, Sep 22 2025
STATUS
approved