OFFSET
1,2
COMMENTS
For a discussion and guide to related arrays, see A208510.
Also the number of multisets of size k that fit within some normal multiset of size n. A multiset is normal if it spans an initial interval of positive integers. - Andrew Howroyd, Sep 18 2018
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
FORMULA
u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = Sum_{i=1..k} binomial(k-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018
T(n,k) = (n - k + 1)*hypergeom([1 - k, n - k + 2], [2], -1). - Peter Luschny, Sep 18 2018
EXAMPLE
Triangle begins:
1;
2, 2;
3, 5, 4;
4, 9, 12, 8;
5, 14, 25, 28, 16;
6, 20, 44, 66, 64, 32;
7, 27, 70, 129, 168, 144, 64;
...
First three polynomials v(n,x): 1, 2 + 2x , 3 + 5x + 4x^2.
The T(3, 1) = 3 multisets: (1), (2), (3).
The T(3, 2) = 5 multisets: (11), (12), (13), (22), (23).
The T(3, 3) = 4 multisets: (111), (112), (122), (123).
MAPLE
T := (n, k) -> simplify((n + 1 - k)*hypergeom([1 - k, -k + n + 2], [2], -1)):
seq(seq(T(n, k), k=1..n), n=1..10); # Peter Luschny, Sep 18 2018
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A208341 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210554 *)
PROG
(PARI) T(n, k)={sum(i=1, k, binomial(k-1, i-1)*binomial(n-k+i, i))} \\ Andrew Howroyd, Sep 18 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 22 2012
EXTENSIONS
Example corrected by Philippe Deléham, Mar 23 2012
STATUS
approved