A363732 Triangle read by rows. The triangle algorithm applied to (-1)^n/n!.

1, -2, 1, 5, -4, 1, -15, 15, -6, 1, 52, -60, 30, -8, 1, -203, 260, -150, 50, -10, 1, 877, -1218, 780, -300, 75, -12, 1, -4140, 6139, -4263, 1820, -525, 105, -14, 1, 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1, -115975, 190323, -149040, 73668, -25578, 6552, -1260, 180, -18, 1

Offset: 0

Author

Peter Luschny, Jun 18 2023

Keywords

• sign
• tabl

Comments

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.

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

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

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.

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.

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.

.

The paradigmatic examples are:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(For Chen's sequence see A363524.)

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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.

Examples

			The triangle T(n, k) starts:
  [0]     1;
  [1]    -2,      1;
  [2]     5,     -4,     1;
  [3]   -15,     15,    -6,      1;
  [4]    52,    -60,    30,     -8,    1;
  [5]  -203,    260,  -150,     50,  -10,    1;
  [6]   877,  -1218,   780,   -300,   75,  -12,   1;
  [7] -4140,   6139, -4263,   1820, -525,  105, -14,   1;
  [8] 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1;
		

Links

• Peter Luschny, Table of n, a(n) for row 0..100.
• Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
• Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
• Naho Kawasaki and Yasuo Ohno, The triangle algorithm for Bernoulli polynomials, Integers, vol. 23 (2023). (See figure 4.)
• D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Crossrefs

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

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

Programs

Maple

TA := proc(a, n, m, x) option remember; if n = 0 then a(m) else
normal((m + 1)*TA(a, n - 1, m + 1, x) - (m + 1 - x)*TA(a, n - 1, m, x)) fi end:
seq(seq(coeff(TA(n -> (-1)^n/n!, n, 0, x), x, k), k = 0..n), n = 0..10);

Mathematica

(* rows[0..n], n[0..oo] *)
(* row[n]= *)
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
(* columns[1..n], n[0..oo] *)
(* column[n]= *)
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
(* sequence *)
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
(* Detlef Meya, Jun 22 2023 *)

SageMath

def a(n): return (-1)^n / factorial(n)
@cached_function
def p(n, m):
    R = PolynomialRing(QQ, "x")
    if n == 0: return R(a(m))
    return R((m + 1)*p(n - 1, m + 1) - (m + 1 - x)*p(n - 1, m))
for n in range(10): print(p(n, 0).list())