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A238385
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Shifted lower triangular matrix A238363 with a main diagonal of ones.
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16
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1, 1, 1, -1, 2, 1, 2, -3, 3, 1, -6, 8, -6, 4, 1, 24, -30, 20, -10, 5, 1, -120, 144, -90, 40, -15, 6, 1, 720, -840, 504, -210, 70, -21, 7, 1, -5040, 5760, -3360, 1344, -420, 112, -28, 8, 1, 40320, -45360, 25920, -10080, 3024, -756, 168, -36, 9, 1, -362880, 403200, -226800, 86400, -25200, 6048, -1260, 240, -45, 10, 1
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OFFSET
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0,5
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COMMENTS
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Shift A238363 and add a main diagonal of ones to obtain this array. The row polynomials form a special Sheffer sequence of polynomials, an Appell sequence.
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LINKS
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FORMULA
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a(n,k) = (-1)^(n+k-1)*n!/((n-k)*k!) for k<n and a(n,n)=1.
Along the n-th diagonal (n>0) Diag(n,k) = a(n+k,k) = (-1)^(n-1)(n-1)! * A007318(n+k,k).
E.g.f.: (log(1+t)+1)*exp(x*t).
E.g.f. for inverse: exp(x*t)/(log(1+t)+1).
The lowering/annihilation and raising/creation operators for the row polynomials are L=D=d/dx and R=x+1/[(1+D)(1+log(1+D))], i.e., L p(n,x)= n*p(n-1,x) and R p(n,x)= p(n+1,x).
E.g.f. of row sums: (log(1+t)+1)*exp(t). Cf. |row sums-1|=|A002741|.
E.g.f. of unsigned row sums: (-log(1-t)+1)*exp(t). Cf. A002104 + 1.
Let dP = A132440, the infinitesimal generator for the Pascal matrix, I, the identity matrix, and T, this entry's lower triangular matrix, then exp(T-I)=I+dP, i.e., T=I+log(I+dP). Also, ((T-I)_n)^n=0, where (T-I)_n denotes the n X n submatrix, i.e., (T-I)_n is nilpotent of order n. - Tom Copeland, Mar 02 2014
Dividing each subdiagonal by its first element (-1)^(n-1)*(n-1)! yields Pascal's triangle A007318. This is equivalent to multiplying the e.g.f. by exp(t)/(log(1+t)+1). - Tom Copeland, Apr 16 2014
B) = [St1]*[dP]*[St1]^(-1) + I
C) = [St2]^(-1)*[dP]*[St2] + I
D) = [St2]^(-1)*[dP]*[St1]^(-1) + I,
p_n(x) = (1 + log(1+D)) x^n = (1 + D - D^2/2 + D^3/3- ...) x^n = x^n + n! * Sum_(k=1,..,n) (-1)^(k+1) (1/k) x^(n-k)/(n-k)!.
Unsigned T with the first two diagonals nulled gives an exponential infinitesimal generator M (infinigen) for the rencontres numbers A008290, and negated M gives the infinigen for A055137; i.e., with M = |T| - I - dP = -log(I-dP)-dP, then e^M = e^(-dP) / (I-dP) = lower triangular A008290, and e^(-M) = e^dP (I-dP) = A007318 * (I-dP) = lower triangular A055137. The matrix formulation is consistent with the operator relations e^(-D) / (1-D) x^n = n-th row polynomial of A008290 and e^D (1-D) x^n = n-th row polynomial of A055137. (End)
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EXAMPLE
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The triangle a(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 1 1
2: -1 2 1
3: 2 -3 3 1
4: -6 8 -6 4 1
5: 24 -30 20 -10 5 1
6: -120 144 -90 40 -15 6 1
7: 720 -840 504 -210 70 -21 7 1
8: -5040 5760 -3360 1344 -420 112 -28 8 1
9: 40320 -45360 25920 -10080 3024 -756 168 -36 9 1
10: -362880 403200 -226800 86400 -25200 6048 -1260 240 -45 10 1
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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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