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A210554 Triangle of coefficients of polynomials v(n,x) jointly generated with A208341; see the Formula section. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)