A363732 - OEIS

%I #21 Jul 30 2023 19:49:27

%S 1,-2,1,5,-4,1,-15,15,-6,1,52,-60,30,-8,1,-203,260,-150,50,-10,1,877,

%T -1218,780,-300,75,-12,1,-4140,6139,-4263,1820,-525,105,-14,1,21147,

%U -33120,24556,-11368,3640,-840,140,-16,1,-115975,190323,-149040,73668,-25578,6552,-1260,180,-18,1

%N Triangle read by rows. The triangle algorithm applied to (-1)^n/n!.

%C The triangle algorithm, as understood here, is a transformation that maps a sequence of integers (a(n) : n >= 0) to a polynomial sequence. A polynomial sequence is a sequence of polynomials (P(n,x) : n >= 0) with degree(P(n, x)) = n for all n >= 0.

%C The polynomials P(n, x) are recursively defined by P(n, x) = p(n, 0, x), where the initial sequence is p(0, m, x) = a(m), and for n > 0 is given by

%C   p(n, m, x) = (m + 1)*p(n - 1, m + 1, x) - (m + 1 - x)*p(n - 1, m, x).

%C Here we identify the polynomial sequence with the infinite lower triangular array of its coefficients, T(n, k) = [x^k] P(n, x). We call the mapping (a(n) : n >= 0) -> (T(n, k) : 0 <= k <= n) the 'triangle algorithm', following the lead of Kawasaki and Ohno.

%C Evaluating P(n, x) at different values of x gives rise to a multitude of other sequences; in particular, the transformation a(n) -> b(n) = P(n, 1) will be called the Akiyama-Tanigawa transform of a.

%C The triangle algorithm was studied by Akiyama and Tanigawa, Chen, Imatomi, Arakawa and Kaneko, Kawasaki and Ohno, and others, at first in connection with the Bernoulli and Poly-Bernoulli numbers.

%C .

%C The paradigmatic examples are:

%C   a(n) = 1 -> x^n, the standard base of polynomials, A023531.

%C   a(n) = n + 1 -> binomial(n, k), Pascal triangle, A007318.

%C   a(n) = n + 1 -> P(n, 1) powers of 2, A000079.

%C   a(n) = n + 1 -> P(n, 0) the all 1's sequence A000012.

%C   a(n) = 2^n -> [x^k] P(n, x), A154921.

%C   a(n) = 2^n -> P(n, 0) Fubini numbers, A000670.

%C   a(n) = 2^n -> P(n, 1) ordered set partitions of subsets of [n], A000629.

%C   a(n) = 2^n -> P(n,-1) osp. of [n] with even number of blocks, A052841.

%C   a(n) = 1 / (n + 1) -> [x^k] B(n, x), Bernoulli polynomials, A196838/A196839.

%C   a(n) = 1 / (n + 1) -> B(n, 1), the Bernoulli numbers, A164555/A027642.

%C   a(n) = Chen(n) -> skp(n, x), Swiss-Knife polynomials, A153641.

%C   a(n) = Chen(n) -> P(n, 0), 2^n*Euler(n, 1/2) = Euler(n), A122045.

%C   a(n) = Chen(n) -> P(n, 1), 2^n*Euler(n, 1), A155585.

%C   a(n) = (-1)^n/n! -> [x^k] P(n, x) this "Bell" triangle.

%C   a(n) = (-1)^n/n! -> (-1)^n*P(n, 1) = Bell(n), A000110.

%C   a(n) = (-1)^n/n! -> (-1)^n*P(n,-1) = 2-Bell(n), A005493.

%C   a(n) = 1/n! -> (-1)^n*P(n, 1) = complementary Bell(n), A000587.

%C   a(n) = 1/n! -> (-1)^n*P(n,-1) = complementary 2-Bell(n), A074051.

%C   (For Chen's sequence see A363524.)

%C .

%C The present sequence deals with the case of the Bell numbers. In contrast to Aitken's array A011971 and its variants A123346 and A011972, the Bell numbers do not appear as a column of the triangle but as row sums (times (-1)^n), i.e., as values of the associated polynomials at x = 1. Comparing this with a similar situation with the Bernoulli numbers/polynomials, our triangle could be viewed as a more organic generalization of the Bell numbers. Indeed, the names 'Bell triangle' and 'Bell polynomials' would be justified here; but these are already assigned to other concepts.

%H Peter Luschny, <a href="/A363732/b363732.txt">Table of n, a(n) for row 0..100</a>.

%H Kwang-Wu Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/CHEN/AlgBE2.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.

%H Masanobu Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.

%H Naho Kawasaki and Yasuo Ohno, <a href="http://math.colgate.edu/~integers/x39/x39.pdf">The triangle algorithm for Bernoulli polynomials</a>, Integers, vol. 23 (2023). (See figure 4.)

%H D. Merlini, R. Sprugnoli, and M. C. Verri, <a href="http://www.emis.de/journals/INTEGERS/papers/f5/f5.Abstract.html">The Akiyama-Tanigawa Transformation</a>, Integers, 5 (1) (2005) #A05.

%e The triangle T(n, k) starts:

%e   [0]     1;

%e   [1]    -2,      1;

%e   [2]     5,     -4,     1;

%e   [3]   -15,     15,    -6,      1;

%e   [4]    52,    -60,    30,     -8,    1;

%e   [5]  -203,    260,  -150,     50,  -10,    1;

%e   [6]   877,  -1218,   780,   -300,   75,  -12,   1;

%e   [7] -4140,   6139, -4263,   1820, -525,  105, -14,   1;

%e   [8] 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1;

%p TA := proc(a, n, m, x) option remember; if n = 0 then a(m) else

%p normal((m + 1)*TA(a, n - 1, m + 1, x) - (m + 1 - x)*TA(a, n - 1, m, x)) fi end:

%p seq(seq(coeff(TA(n -> (-1)^n/n!, n, 0, x), x, k), k = 0..n), n = 0..10);

%t (* rows[0..n], n[0..oo] *)

%t (* row[n]= *)

%t n=9;r={};For[a=n+1,a>0,a--,AppendTo[r,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j))/(2*j+2),{j,0,n-a}]]];r

%t (* columns[1..n], n[0..oo] *)

%t (* column[n]= *)

%t n=0;c={};For[a=1,a<15,a++,AppendTo[c,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j-1))/(2*j),{j,1,n}]]];c

%t (* sequence *)

%t s={};For[n=0,n<15,n++,For[a=n+1,a>0,a--,AppendTo[s,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j))/(2*j+2),{j,0,n-a}]]]];s

%t (* _Detlef Meya_, Jun 22 2023 *)

%o (SageMath)

%o def a(n): return (-1)^n / factorial(n)

%o @cached_function

%o def p(n, m):

%o     R = PolynomialRing(QQ, "x")

%o     if n == 0: return R(a(m))

%o     return R((m + 1)*p(n - 1, m + 1) - (m + 1 - x)*p(n - 1, m))

%o for n in range(10): print(p(n, 0).list())

%Y Cf. A293037 (row sums), A000110 (row sums, unsigned), A005493 (alternating row sums, signed).

%Y Cf. A023531, A007318, A000079, A000012, A154921, A000670, A196838/A196839, A164555/A027642, A153641, A122045, A155585, A011971, A123346, A011972, A056857, A363524 (Chen sequence).

%K sign,tabl

%O 0,2

%A _Peter Luschny_, Jun 18 2023