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A079641 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.

%I A079641 #52 Dec 16 2024 08:53:10
%S A079641 1,2,1,6,6,1,26,36,12,1,150,250,120,20,1,1082,2040,1230,300,30,1,9366,
%T A079641 19334,13650,4270,630,42,1,94586,209580,166376,62160,11900,1176,56,1,
%U A079641 1091670,2562354,2229444,952728,220500,28476,2016,72,1,14174522
%N A079641 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.
%C A079641 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
%C A079641 Subtriangle of triangle in A129062. - _Philippe Deléham_, Feb 17 2013
%C A079641 Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%H A079641 Nick Early, <a href="https://arxiv.org/abs/1712.08520">Canonical Bases for Permutohedral Plates</a>, arXiv:1712.08520 [math.CO], 2017.
%H A079641 Nick Early, <a href="https://arxiv.org/abs/1810.03246">Honeycomb tessellations and canonical bases for permutohedral blades</a>, arXiv:1810.03246 [math.CO], 2018.
%H A079641 D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, arXiv:math/9207221 [math.CA], 1992; The Mathematica J., 2 (1992), 67-78.
%F A079641 T(n, k) = Sum_{i=k..n} A008277(n, i) * |A008275(i, k)|.
%F A079641 E.g.f.: (2-exp(x))^(-y). - _Vladeta Jovovic_, Nov 22 2003
%F A079641 From _Peter Bala_, Sep 12 2011: (Start)
%F A079641 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].
%F A079641 There is a Dobinski-type formula: exp(-x)*Sum_{k >= 0} (-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).
%F A079641 The row polynomials also satisfy an analog of the Bernoulli's summation formula for powers of integers, namely, 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)
%F A079641 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
%F A079641 Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x*z/(1 - 2*z/(1 - (x + 1)*z/(1 - 4*z/(1 - (x + 2)*z/(1 - 6*z/(1 - (x + 3)*z/(1 - 8*z/(1 - ... ))))))))) =  1 + x*z + (2*x + x^2)*z^2 + (6*x + 6*x^2 + x^3)*z^3 + .... - _Peter Bala_, Dec 12 2024
%e A079641 Triangle begins:
%e A079641   1;
%e A079641   2,1;
%e A079641   6,6,1;
%e A079641   26,36,12,1;
%e A079641   150,250,120,20,1;
%e A079641   1082,2040,1230,300,30,1;
%e A079641   ...
%e A079641 Triangle (0,2,1,4,2,6,3,8,4,...) DELTA (1,0,1,0,1,0,1,0,1,...) begins:
%e A079641   1
%e A079641   0, 1
%e A079641   0, 2, 1
%e A079641   0, 6, 6, 1
%e A079641   0, 26, 36, 12, 1
%e A079641   0, 150, 250, 120, 20, 1
%e A079641   0, 1082, 2040, 1230, 300, 30, 1. - _Philippe Deléham_, Dec 22 2011
%p A079641 # The function BellMatrix is defined in A264428.
%p A079641 # Adds (1, 0, 0, 0, ..) as column 0.
%p A079641 BellMatrix(n -> add((-1)^(n-k)*2^k*k!*combinat:-stirling2(n, k), k=0..n), 9); # _Peter Luschny_, Jan 26 2016
%t A079641 rows = 10;
%t A079641 t = Table[Sum[(-1)^(n-k)*2^k*k!*StirlingS2[n, k], {k,0,n}], {n, 0, rows}];
%t A079641 T[n_, k_] := BellY[n, k, t];
%t A079641 Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%Y A079641 Cf. A000670 (row sums), A000629 (first column), A195204, A195205. A209849, A129062
%K A079641 nonn,tabl
%O A079641 1,2
%A A079641 _Vladeta Jovovic_, Jan 30 2003