A168296 Worpitzky form polynomials for the {1,16,1} A142462 sequence: p(x,n) = Sum_{k=1..n} A(n, k)*binomial(x + k - 1, n - 1).

1, 1, 2, 2, 18, 18, 6, 156, 432, 288, 24, 792, 7416, 13248, 6624, 120, -11280, 64800, 374400, 496800, 198720, 720, -62640, -1254960, 4968000, 20865600, 22057920, 7352640, 5040, 24012000, -11854080, -125677440, 389491200, 1288103040

Offset

1,3

Comments

Row sums: {1, 3, 38, 882, 28104, 1123560, 53927280, 3019902480, 193273557120, 13915694298240, ...}.

In Comtet there is this function:

x^n = Sum_{k=1..n} Eulerian(n, k*binomial(x+k-1, n)

In the OEIS I was looking for an Umbral Calculus expansion for the MacMahon and found this "Worpitzky form":

Sum_{k=1..n} MacMahon(n,k)*binomial(x+k-1, n-1) = (2*x+1)^(n+1)

They use the infinite sums k, 2*k+1 type polynomials and are pretty much alike except for a sliding offset in n.

Conjecture: "Worpitzky forms"

Some general polynomial form: general Pascal recursion Pascal(n,k,m)

p(x,n,m) = Sum_{k=1..n} Pascal(n,k,m)*binomial(x+k-1,n-1)

where p(x,n,m) are the inverse z transform polynomials.

Links

Table of n, a(n) for n=1..34.

Formula

p(x,n) = Sum_{k=1..n} A(n, k)*binomial(x + k - 1, n - 1).

Example

{1},
{1, 2},
{2, 18, 18},
{6, 156, 432, 288},
{24, 792, 7416, 13248, 6624},
{120, -11280, 64800, 374400, 496800, 198720},
{720, -62640, -1254960, 4968000, 20865600, 22057920, 7352640},
{5040, 24012000, -11854080, -125677440, 389491200, 1288103040, 1132306560, 323516160},
{40320, 192378240, 5004581760, -1669248000, -12569437440, 32116331520, 87702289920, 65997296640, 16499324160},
{362880, -119545632000, 57161064960, 868954106880, -218287560960, -1293900894720, 2812649495040, 6545378949120, 4306323605760, 956960801280}

Mathematica

(* Worpitzky form polynomials for A142462 *)
Clear[A, m, n, k, a, p]
m = 7;
A[n_, 1] := 1 A[n_, n_] := 1
A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k];
a = Table[A[n, k], {n, 10}, {k, n}];
p[x_, n_] = Sum[a[[n, k]]*Binomial[x + k - 1, n - 1], {k, 1, n}];
Table[CoefficientList[Expand[(n - 1)!*p[x, n]], x], {n, 1, 10}];
Flatten[%]

Crossrefs

Cf. A142462.

Sequence in context: A225123 A087338 A055735 * A205454 A100304 A213271

Adjacent sequences: A168293 A168294 A168295 * A168297 A168298 A168299

Keyword

uned,sign

Author

Roger L. Bagula, Nov 22 2009

Status

approved