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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

Table of n, a(n) for n=0..35.

Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.

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}];

Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2019, from Maple *)

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

Row sums are A000007, alternating row sums are A210657.

Cf. T(n,n) = A000680, T(n, 0) = A028296(n) (Gudermannian), A000364 (Euler secant), A241171 (Joffe's differences), A028246 (Worpitzky).

Cf. A167374 (m=0), A028246 & A163626 (m=1), this seq (m=2), A318260 (m=3).

Sequence in context: A077806 A019305 A274266 * A304935 A156274 A079778

Adjacent sequences:  A318256 A318257 A318258 * A318260 A318261 A318262

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Sep 06 2018

STATUS

approved

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Last modified May 13 19:36 EDT 2021. Contains 343868 sequences. (Running on oeis4.)