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A227343
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Matrix inverse of triangle A227342.
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2
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1, 1, 1, 3, 3, 1, 13, 13, 6, 1, 75, 75, 37, 10, 1, 541, 541, 270, 85, 15, 1, 4683, 4683, 2341, 770, 170, 21, 1, 47293, 47293, 23646, 7861, 1890, 308, 28, 1, 545835, 545835, 272917, 90930, 22491, 4158, 518, 36, 1, 7087261, 7087261, 3543630, 1181125, 294525, 57351, 8400, 822, 45, 1
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OFFSET
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0,4
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COMMENTS
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The e.g.f. has the form A(t)*exp(x*B(t)), where A(t) = 1/(2 - exp(t)) and B(t) = exp(t) - 1. Thus the row polynomials of this triangle form a Sheffer sequence for the pair (1 - t, log(1 + t)) (see Roman, p.17).
Let x_(k) := x*(x-1)*...*(x-k+1) denote the k-th falling factorial polynomial. Define a sequence x_[n] of basis polynomials for the polynomial algebra C[x] by setting x_[0] = 1, and setting x_[n] = x_(n-1)*(x - 2*n + 1) for n >= 1. The sequence begins [1, x-1, x*(x-3), x*(x-1)*(x-5), x*(x-1)*(x-2)*(x-7), ...]. Then this is the triangle of connection constants for expressing the monomial polynomials x^n as a linear combination of the basis x_[k], that is, x^n = sum {k = 0..n} T(n,k)*x_[k]. An example is given below.
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REFERENCES
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S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.
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LINKS
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FORMULA
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E.g.f.: 1/(2 - exp(t))*exp(x*(exp(t) - 1)) = 1 + (1 + x)*t + (3 + 3*x + x^2)*t^2/2! + (13 + 13*x + 6*x^2 + x^3)*t^3/3! + ....
Recurrence equation: T(n,0) = A000670(n), and for k >= 1, T(n,k) = 1/k*sum {i = 1..n} binomial(n,i)*T(n-i,k-1).
The row polynomials R(n,x) satisfy the Sheffer identity R(n,x + y) = sum {k = 0..n} binomial(n,k)*Bell(k,y)*R(n-k,x), where Bell(k,y) is the Bell or exponential polynomial (row polynomials of A048993).
The row polynomials also satisfy d/dx(R(n,x)) = sum {k = 0..n-1} binomial(n,k)*R(k,x).
Row sums A059099. Column 1 and column 2 = A000670. 1 + 2*column 3 = A000670 (apart from the first two terms).
T(n,k) = (n!/k!)*[t^n](1/(2-exp(t))(exp(t)-1)^k.
T(n,k) = (n!/k!)*[t^(n-k)](t/log(1+t))^(n+1)/(1-t^2). (End)
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EXAMPLE
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Triangle begins
n\k| 0 1 2 3 4 5
= = = = = = = = = = = = = = = = =
0 | 1
1 | 1 1
2 | 3 3 1
3 | 13 13 6 1
4 | 75 75 37 10 1
5 | 541 541 270 85 15 1
...
Connection constants. Row 4 = [75,75,37,10,1]: Thus
75 + 75*(x - 1) + 37*x*(x - 3) + 10*x*(x - 1)*(x - 5)+ x*(x - 1)*(x - 2)*(x - 7) = x^4.
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MATHEMATICA
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T[n_, k_] := n!/k! SeriesCoefficient[Series[1/(2 - Exp[t]) (Exp[t] - 1)^k, {t, 0, n}], n]
Flatten[Table[T[n, k], {n, 0, 12}, {k, 0, n}]]
U[n_, k_] := n!/k! SeriesCoefficient[Series[1/(1 - t^2) (t/Log[1 + t])^(n + 1), {t, 0, n - k}], n - k]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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