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%I A363732 #21 Jul 30 2023 19:49:27
%S A363732 1,-2,1,5,-4,1,-15,15,-6,1,52,-60,30,-8,1,-203,260,-150,50,-10,1,877,
%T A363732 -1218,780,-300,75,-12,1,-4140,6139,-4263,1820,-525,105,-14,1,21147,
%U A363732 -33120,24556,-11368,3640,-840,140,-16,1,-115975,190323,-149040,73668,-25578,6552,-1260,180,-18,1
%N A363732 Triangle read by rows. The triangle algorithm applied to (-1)^n/n!.
%C A363732 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 A363732 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 A363732 p(n, m, x) = (m + 1)*p(n - 1, m + 1, x) - (m + 1 - x)*p(n - 1, m, x).
%C A363732 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 A363732 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 A363732 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 A363732 .
%C A363732 The paradigmatic examples are:
%C A363732 a(n) = 1 -> x^n, the standard base of polynomials, A023531.
%C A363732 a(n) = n + 1 -> binomial(n, k), Pascal triangle, A007318.
%C A363732 a(n) = n + 1 -> P(n, 1) powers of 2, A000079.
%C A363732 a(n) = n + 1 -> P(n, 0) the all 1's sequence A000012.
%C A363732 a(n) = 2^n -> [x^k] P(n, x), A154921.
%C A363732 a(n) = 2^n -> P(n, 0) Fubini numbers, A000670.
%C A363732 a(n) = 2^n -> P(n, 1) ordered set partitions of subsets of [n], A000629.
%C A363732 a(n) = 2^n -> P(n,-1) osp. of [n] with even number of blocks, A052841.
%C A363732 a(n) = 1 / (n + 1) -> [x^k] B(n, x), Bernoulli polynomials, A196838/A196839.
%C A363732 a(n) = 1 / (n + 1) -> B(n, 1), the Bernoulli numbers, A164555/A027642.
%C A363732 a(n) = Chen(n) -> skp(n, x), Swiss-Knife polynomials, A153641.
%C A363732 a(n) = Chen(n) -> P(n, 0), 2^n*Euler(n, 1/2) = Euler(n), A122045.
%C A363732 a(n) = Chen(n) -> P(n, 1), 2^n*Euler(n, 1), A155585.
%C A363732 a(n) = (-1)^n/n! -> [x^k] P(n, x) this "Bell" triangle.
%C A363732 a(n) = (-1)^n/n! -> (-1)^n*P(n, 1) = Bell(n), A000110.
%C A363732 a(n) = (-1)^n/n! -> (-1)^n*P(n,-1) = 2-Bell(n), A005493.
%C A363732 a(n) = 1/n! -> (-1)^n*P(n, 1) = complementary Bell(n), A000587.
%C A363732 a(n) = 1/n! -> (-1)^n*P(n,-1) = complementary 2-Bell(n), A074051.
%C A363732 (For Chen's sequence see A363524.)
%C A363732 .
%C A363732 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 A363732 Peter Luschny, <a href="/A363732/b363732.txt">Table of n, a(n) for row 0..100</a>.
%H A363732 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 A363732 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 A363732 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 A363732 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 A363732 The triangle T(n, k) starts:
%e A363732 [0] 1;
%e A363732 [1] -2, 1;
%e A363732 [2] 5, -4, 1;
%e A363732 [3] -15, 15, -6, 1;
%e A363732 [4] 52, -60, 30, -8, 1;
%e A363732 [5] -203, 260, -150, 50, -10, 1;
%e A363732 [6] 877, -1218, 780, -300, 75, -12, 1;
%e A363732 [7] -4140, 6139, -4263, 1820, -525, 105, -14, 1;
%e A363732 [8] 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1;
%p A363732 TA := proc(a, n, m, x) option remember; if n = 0 then a(m) else
%p A363732 normal((m + 1)*TA(a, n - 1, m + 1, x) - (m + 1 - x)*TA(a, n - 1, m, x)) fi end:
%p A363732 seq(seq(coeff(TA(n -> (-1)^n/n!, n, 0, x), x, k), k = 0..n), n = 0..10);
%t A363732 (* rows[0..n], n[0..oo] *)
%t A363732 (* row[n]= *)
%t A363732 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 A363732 (* columns[1..n], n[0..oo] *)
%t A363732 (* column[n]= *)
%t A363732 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 A363732 (* sequence *)
%t A363732 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 A363732 (* _Detlef Meya_, Jun 22 2023 *)
%o A363732 (SageMath)
%o A363732 def a(n): return (-1)^n / factorial(n)
%o A363732 @cached_function
%o A363732 def p(n, m):
%o A363732 R = PolynomialRing(QQ, "x")
%o A363732 if n == 0: return R(a(m))
%o A363732 return R((m + 1)*p(n - 1, m + 1) - (m + 1 - x)*p(n - 1, m))
%o A363732 for n in range(10): print(p(n, 0).list())
%Y A363732 Cf. A293037 (row sums), A000110 (row sums, unsigned), A005493 (alternating row sums, signed).
%Y A363732 Cf. A023531, A007318, A000079, A000012, A154921, A000670, A196838/A196839, A164555/A027642, A153641, A122045, A155585, A011971, A123346, A011972, A056857, A363524 (Chen sequence).
%K A363732 sign,tabl
%O A363732 0,2
%A A363732 _Peter Luschny_, Jun 18 2023