OFFSET
1,2
COMMENTS
Triangle T(n,k), 1<=k<=n, read by rows, given by (0, 2, 1, 4, 2, 6, 3, 8, 4, 10, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011
Subtriangle of triangle in A129062. - Philippe Deléham, Feb 17 2013
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
LINKS
Nick Early, Canonical Bases for Permutohedral Plates, arXiv:1712.08520 [math.CO], 2017.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; The Mathematica J., 2 (1992), 67-78.
FORMULA
E.g.f.: (2-exp(x))^(-y). - Vladeta Jovovic, Nov 22 2003
From Peter Bala, Sep 12 2011: (Start)
The row generating polynomials R(n,x) begin R(1,x) = x, R(2,x) = 2*x+x^2, R(3,x) = 6*x+6*x^2+x^3 and satisfy the recurrence R(n+1,x) = x*(2*R(n,x+1) - R(n,x)). They form a sequence of binomial type polynomials. In particular, denoting R(n,x) by x^[n] to emphasize the analogies with the monomial polynomials x^n, we have the binomial expansion (x+y)^[n] = sum {k = 0..n} binomial(n,k)*x^[n-k]*y^[k].
There is a Dobinski-type formula exp(-x)*sum {k = 0..inf} (-k)^[n]*x^k/k! = Bell(n,-x). The alternating n-th row entries (-1)^k*T(n,k) are the connection coefficients expressing the polynomial Bell(n,-x) as a linear combination of Bell(k,x), 1<=k<=n. For example, the list of coefficients of R(4,x) is [26,36,12,1] and we have Bell(4,-x) = -26*Bell(1,x) + 36*Bell(2,x) - 12*Bell(3,x) + Bell(4,x).
The row polynomials also satisfy an analog of the Bernoulli's summation formula for powers of integers sum {k = 1..n} k^[p] = 1/(p+1) sum {k = 0..p} binomial(p+1,k) * B_k * n^[p+1-k], where B_k denotes the Bernoulli numbers. Compare with A195204 and A195205.
(End)
Let D be the forward difference operator D(f(x)) = f(x+1) - f(x). Then the n-th row polynomial R(n,x) = 1/f(x)*(x*D)^n(f(x)) with f(x) = 2^x. Cf. A209849. Also cf. A008277 where the row polynomials are given by 1/f(x)*(x*d/dx)^n(f(x))), where now f(x) = exp(x). - Peter Bala, Mar 16 2012
EXAMPLE
Triangle begins:
1;
2,1;
6,6,1;
26,36,12,1;
150,250,120,20,1;
1082,2040,1230,300,30,1;
...
Triangle (0,2,1,4,2,6,3,8,4,...) DELTA (1,0,1,0,1,0,1,0,1,...) begins:
1
0, 1
0, 2, 1
0, 6, 6, 1
0, 26, 36, 12, 1
0, 150, 250, 120, 20, 1
0, 1082, 2040, 1230, 300, 30, 1. - Philippe Deléham, Dec 22 2011
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add((-1)^(n-k)*2^k*k!*combinat:-stirling2(n, k), k=0..n), 9); # Peter Luschny, Jan 26 2016
MATHEMATICA
rows = 10;
t = Table[Sum[(-1)^(n-k)*2^k*k!*StirlingS2[n, k], {k, 0, n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladeta Jovovic, Jan 30 2003
STATUS
approved