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

1,2

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

Row sums are A027941.

Cf. A160232, A208341, A208510, A303974.

Sequence in context: A317050 A243970 A282443 * A208912 A210212 A209762

Adjacent sequences: A210551 A210552 A210553 * A210555 A210556 A210557

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 22 2012

Extensions

Example corrected by Philippe Deléham, Mar 23 2012

Status

approved