This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A329070 #69 Feb 16 2025 08:33:58
%S A329070 1,1,1,1,2,1,1,8,6,1,1,48,180,24,1,1,384,12960,8064,120,1,1,3840,
%T A329070 1710720,10644480,604800,720,1,1,46080,359251200,35765452800,
%U A329070 19813248000,68428800,5040,1,1,645120,109930867200,244635697152000,2303884477440000,70355755008000,10897286400,40320,1
%N A329070 Array read by ascending antidiagonals: T(n, k) = (k*n)!/(k^n*(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol.
%C A329070 For information about the function W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)) (mentioned in the Formula section below), see Theorem 3.2 in Elizalde and Noy (2003) with u = 0 and m and a in the theorem equal to our m + 1. See also the documentation of array A327722.
%C A329070 By using the ratio test and the Stirling approximation to the gamma function, we may show that the radius of convergence of the power series for W_m(z) is infinity (for each m >= 0). Thus, the function W_m(z) (as defined by the above power series) is entire.
%C A329070 If we define S(m,s) = T(n-s, s+1) for m >= 0 and 0 <= s <= m, we get the triangular array that appears in the Example section below.
%H A329070 Sergi Elizalde and Marc Noy, <a href="https://doi.org/10.1016/S0196-8858(02)00527-4 ">Consecutive patterns in permutations</a>, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116).
%H A329070 Alison Schuetz and Gwyneth Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arXiv:1401.7194 [math.CO], 2014.
%H A329070 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%H A329070 Wikipedia, <a href="https://en.wikipedia.org/wiki/Falling_and_rising_factorials">Falling and rising factorials</a>.
%F A329070 T(0,k) = 1, T(1,k) = k!, and T(2,k) = (2*k)!/(k + 1) for k >= 1.
%F A329070 T(n,1) = 1, T(n,2) = (2*n)!!, and T(n,3) is related to the Airy functions (see the documentation of A176730).
%F A329070 T(n+1,k) = (k-1)! * binomial(k*(n+1), k-1) * T(n,k) for n >= 0 and k >= 1.
%F A329070 T(n+1,k)/(k! * T(n,k)) = Cat(n+1, k), where Cat(d, k) = binomial(k*d, k)/(k * (d - 1) + 1) is a Fuss-Catalan number; see Theorem 1.2 in Schuetz and Whieldon (2014).
%F A329070 If F(k,z) = Sum_{n >= 0} z^(k*n)/T(n,k), then F(k,z) satisfies the o.d.e. F^(k-1)(k,z) - z*F(k,z) = 0.
%F A329070 If W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)), then 1/W_m(z) is the e.g.f. of row m of A327722(m,n), which counts permutations of [n] that avoid the consecutive pattern 12...(m+1)(m+3)(m+2) (or equivalently, the consecutive pattern (m+3)(m+2)...(3)(1)(2)).
%F A329070 The function W_m(z) satisfies the o.d.e. W_m^(m+2)(z) + z*W_m'(z) = 0 with W_m(0) = 1, W_m'(0) = -1, and W_m^(s)(0) = 0 for s = 2..(m + 1).
%e A329070 Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
%e A329070 1, 1, 1, 1, 1, 1, ...
%e A329070 1, 2, 6, 24, 120, 720, ...
%e A329070 1, 8, 180, 8064, 604800, 68428800, ...
%e A329070 1, 48, 12960, 10644480, 19813248000, 70355755008000, ...
%e A329070 ...
%e A329070 Triangular array S(m,s) = T(m-s, s+1) (with rows m >= 0 and columns s >= 0):
%e A329070 1;
%e A329070 1, 1;
%e A329070 1, 2, 1;
%e A329070 1, 8, 6, 1;
%e A329070 1, 48, 180, 24, 1;
%e A329070 1, 384, 12960, 8064, 120, 1;
%e A329070 1, 3840, 1710720, 10644480, 604800, 720, 1;
%e A329070 1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1;
%e A329070 ...
%p A329070 A := (n, k) -> `if`(k=0, 1, (GAMMA(1/k)*GAMMA(k*n+1))/(GAMMA(n+1/k)*k^n)):
%p A329070 seq(seq(A(n-k-1, k), k=1..n-1), n=0..10); # _Peter Luschny_, Nov 04 2019
%Y A329070 Rows include A000012 (n = 0), A000142 (n = 1), A060593 (n = 2).
%Y A329070 Columns include A000012 (k = 1), A000165 (k = 2), A176730 (k = 3).
%Y A329070 Ratios T(n+1,k)/(k!*T(n,k)) include A000012 (k = 1), A000027 (k = 2), A000326 (k = 3), A100157 (k = 4), A234043 (k = 5).
%Y A329070 Cf. A111004, A117226, A202213, A327722.
%K A329070 nonn,tabl
%O A329070 0,5
%A A329070 _Petros Hadjicostas_, Nov 03 2019