A210554 Triangle of coefficients of polynomials v(n,x) jointly generated with A208341; see the Formula section.

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, 8, 35, 104, 225, 360, 416, 320, 128, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512

Offset: 1

Author

Clark Kimberling, Mar 22 2012

Keywords

• nonn
• tabl

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

Examples

			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).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Row sums are A027941.

Cf. A160232, A208341, A208510, A303974.

Programs

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 *)

PARI

T(n,k)={sum(i=1, k, binomial(k-1, i-1)*binomial(n-k+i, i))} \\ Andrew Howroyd, Sep 18 2018

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

Extensions

Example corrected by Philippe Deléham, Mar 23 2012