|
|
A318259
|
|
Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.
|
|
1
|
|
|
1, -1, 1, 5, -11, 6, -61, 211, -240, 90, 1385, -6551, 11466, -8820, 2520, -50521, 303271, -719580, 844830, -491400, 113400, 2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400, -199360981, 1704396331, -6187282920, 12372329970, -14727913200, 10443232800, -4086482400, 681080400
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See the cross-references for some other members.
The unsigned numbers have row sums A210657 which points to an interpretation of the unsigned numbers as a refinement of marked Schröder paths (see Josuat-Vergès and Kim).
|
|
LINKS
|
|
|
FORMULA
|
Let S(n, k) denote Joffe's central differences of zero (A241171) extended to the case n = 0 and k = 0 by prepending a column 1, 0, 0, 0,... to the triangle, then:
T(n,k) = Sum_{j=0..k}((-1)^(k-j)*C(n-j,n-k)*Sum_{i=0..n}((-1)^i*S(n,i)*C(n-i,j))).
|
|
EXAMPLE
|
[0] [ 1]
[1] [ -1, 1]
[2] [ 5, -11, 6]
[3] [ -61, 211, -240, 90]
[4] [ 1385, -6551, 11466, -8820, 2520]
[5] [ -50521, 303271, -719580, 844830, -491400, 113400]
[6] [2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400]
|
|
MAPLE
|
Joffe := proc(n, k) option remember; if k > n then 0 elif k = 0 then k^n else
k*(2*k-1)*Joffe(n-1, k-1)+k^2*Joffe(n-1, k) fi end:
T := (n, k) -> add((-1)^(k-j)*binomial(n-j, n-k)*add((-1)^i*Joffe(n, i)*
binomial(n-i, j), i=0..n), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..6);
|
|
MATHEMATICA
|
Joffe[0, 0] = 1; Joffe[n_, k_] := Joffe[n, k] = If[k>n, 0, If[k == 0, k^n, k*(2*k-1)*Joffe[n-1, k-1] + k^2*Joffe[n-1, k]]];
T[n_, k_] := Sum[(-1)^(k-j)*Binomial[n-j, n-k]*Sum[(-1)^i*Joffe[n, i]* Binomial[n-i, j], {i, 0, n}], {j, 0, k}];
|
|
PROG
|
(Sage)
def EW(m, n):
@cached_function
def S(m, n):
R.<x> = ZZ[]
if n == 0: return R(1)
return R(sum(binomial(m*n, m*k)*S(m, n-k)*x for k in (1..n)))
s = S(m, n).list()
c = lambda k: sum((-1)^(k-j)*binomial(n-j, n-k)*
sum((-1)^i*s[i]*binomial(n-i, j) for i in (0..n)) for j in (0..k))
return [c(k) for k in (0..n)]
def A318259row(n): return EW(2, n)
flatten([A318259row(n) for n in (0..6)])
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|