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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.

%I A318259 #18 Feb 18 2019 06:20:58
%S A318259 1,-1,1,5,-11,6,-61,211,-240,90,1385,-6551,11466,-8820,2520,-50521,
%T A318259 303271,-719580,844830,-491400,113400,2702765,-19665491,58998126,
%U A318259 -93511440,82661040,-38669400,7484400,-199360981,1704396331,-6187282920,12372329970,-14727913200,10443232800,-4086482400,681080400
%N A318259 Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.
%C A318259 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 A318259 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 A318259 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 A318259 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 A318259 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 A318259 [0] [      1]
%e A318259 [1] [     -1,         1]
%e A318259 [2] [      5,       -11,        6]
%e A318259 [3] [    -61,       211,     -240,        90]
%e A318259 [4] [   1385,     -6551,    11466,     -8820,     2520]
%e A318259 [5] [ -50521,    303271,  -719580,    844830,  -491400,    113400]
%e A318259 [6] [2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400]
%p A318259 Joffe := proc(n, k) option remember; if k > n then 0 elif k = 0 then k^n else
%p A318259 k*(2*k-1)*Joffe(n-1, k-1)+k^2*Joffe(n-1, k) fi end:
%p A318259 T := (n, k) -> add((-1)^(k-j)*binomial(n-j, n-k)*add((-1)^i*Joffe(n,i)*
%p A318259 binomial(n-i, j), i=0..n), j=0..k):
%p A318259 seq(seq(T(n, k), k=0..n), n=0..6);
%t A318259 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 A318259 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 A318259 Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 18 2019, from Maple *)
%o A318259 (Sage)
%o A318259 def EW(m, n):
%o A318259     @cached_function
%o A318259     def S(m, n):
%o A318259         R.<x> = ZZ[]
%o A318259         if n == 0: return R(1)
%o A318259         return R(sum(binomial(m*n, m*k)*S(m, n-k)*x for k in (1..n)))
%o A318259     s = S(m, n).list()
%o A318259     c = lambda k: sum((-1)^(k-j)*binomial(n-j,n-k)*
%o A318259         sum((-1)^i*s[i]*binomial(n-i,j) for i in (0..n)) for j in (0..k))
%o A318259     return [c(k) for k in (0..n)]
%o A318259 def A318259row(n): return EW(2, n)
%o A318259 flatten([A318259row(n) for n in (0..6)])
%Y A318259 Row sums are A000007, alternating row sums are A210657.
%Y A318259 Cf. T(n,n) = A000680, T(n, 0) = A028296(n) (Gudermannian), A000364 (Euler secant), A241171 (Joffe's differences), A028246 (Worpitzky).
%Y A318259 Cf. A167374 (m=0), A028246 & A163626 (m=1), this seq (m=2), A318260 (m=3).
%K A318259 sign,tabl
%O A318259 0,4
%A A318259 _Peter Luschny_, Sep 06 2018