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A079641
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Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.
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6
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1, 2, 1, 6, 6, 1, 26, 36, 12, 1, 150, 250, 120, 20, 1, 1082, 2040, 1230, 300, 30, 1, 9366, 19334, 13650, 4270, 630, 42, 1, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1, 14174522
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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. - DELEHAM Philippe, Dec 22 2011
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FORMULA
| T(n, k) = Sum_{i=k..n} A008277(n, i) * |A008275(i, k)|.
E.g.f.: (2-exp(x))^(-y). - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 22 2003
From Peter Bala, Sept 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)
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EXAMPLE
| 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 - DELEHAM Philippe, Dec 22 2011
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CROSSREFS
| Cf. A000670 (row sums), A000629 (first column), A195204, A195205.
Sequence in context: A105278 A008297 A090582 * A182729 A075181 A052121
Adjacent sequences: A079638 A079639 A079640 * A079642 A079643 A079644
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KEYWORD
| nonn,tabl
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 30 2003
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