A318259 - OEIS

%I #18 Feb 18 2019 06:20:58

%S 1,-1,1,5,-11,6,-61,211,-240,90,1385,-6551,11466,-8820,2520,-50521,

%T 303271,-719580,844830,-491400,113400,2702765,-19665491,58998126,

%U -93511440,82661040,-38669400,7484400,-199360981,1704396331,-6187282920,12372329970,-14727913200,10443232800,-4086482400,681080400

%N Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.

%C The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See the cross-references for some other members.

%C 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).

%H Matthieu Josuat-Vergès and Jang Soo Kim, <a href="http://arxiv.org/abs/1101.5608">Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity</a>, arXiv:1101.5608 [math.CO], 2011.

%F 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:

%F 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))).

%e [0] [      1]

%e [1] [     -1,         1]

%e [2] [      5,       -11,        6]

%e [3] [    -61,       211,     -240,        90]

%e [4] [   1385,     -6551,    11466,     -8820,     2520]

%e [5] [ -50521,    303271,  -719580,    844830,  -491400,    113400]

%e [6] [2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400]

%p Joffe := proc(n, k) option remember; if k > n then 0 elif k = 0 then k^n else

%p k*(2*k-1)*Joffe(n-1, k-1)+k^2*Joffe(n-1, k) fi end:

%p T := (n, k) -> add((-1)^(k-j)*binomial(n-j, n-k)*add((-1)^i*Joffe(n,i)*

%p binomial(n-i, j), i=0..n), j=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..6);

%t 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 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}];

%t Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 18 2019, from Maple *)

%o (Sage)

%o def EW(m, n):

%o     @cached_function

%o     def S(m, n):

%o         R.<x> = ZZ[]

%o         if n == 0: return R(1)

%o         return R(sum(binomial(m*n, m*k)*S(m, n-k)*x for k in (1..n)))

%o     s = S(m, n).list()

%o     c = lambda k: sum((-1)^(k-j)*binomial(n-j,n-k)*

%o         sum((-1)^i*s[i]*binomial(n-i,j) for i in (0..n)) for j in (0..k))

%o     return [c(k) for k in (0..n)]

%o def A318259row(n): return EW(2, n)

%o flatten([A318259row(n) for n in (0..6)])

%Y Row sums are A000007, alternating row sums are A210657.

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

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

%K sign,tabl

%O 0,4

%A _Peter Luschny_, Sep 06 2018