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A178340
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Triangle T(n,m) read by rows: denominator of the coefficient [x^m] of the umbral inverse Bernoulli polynomial B^{-1}(n,x).
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2
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1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 5, 1, 1, 1, 1, 6, 1, 2, 3, 2, 1, 7, 1, 1, 1, 1, 1, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 10, 1, 2, 1, 1, 5, 1, 1, 2, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 2, 3, 4, 1, 1, 1, 4, 3, 2, 1, 13
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OFFSET
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0,2
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COMMENTS
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This is the triangle of denominators associated with the numerators of A178252.
(Unlike the coefficients of the Bernoulli Polynomials, the coefficients of the umbral inverse Bernoulli polynomials are all positive.)
Usually T(n,m) = A003989(n-m+1,m) for m>=1, but since we are tabulating denominators of reduced fractions here, this formula may be wrong by a cancelling integer factor.
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LINKS
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FORMULA
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T(n,0) = n+1.
Recurrence for the rational triangle
TinvB(n,m):= A178252(n,m) / T(n,m) from the Sheffer a-sequence, which is 1, (repeat 0), see the comment under A178252: TinvB(n,m) = (n/m)*TinvB(n-1,m-1), for n >= m >= 1. From the z-sequence: TinvB(n,0) = n*Sum_{j=0..n-1} z_j * TinvB(n-1,j), n >= 1, TinvB(0,0) = 1. - Wolfdieter Lang, Aug 25 2015
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EXAMPLE
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The triangle T(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
0: 1
1: 2 1
2: 3 1 1
3: 4 1 2 1
4: 5 1 1 1 1
5: 6 1 2 3 2 1
6: 7 1 1 1 1 1 1
7: 8 1 2 1 4 1 2 1
8: 9 1 1 3 1 1 3 1 1
9: 10 1 2 1 1 5 1 1 2 1
10: 11 1 1 1 1 1 1 1 1 1 1
11: 12 1 2 3 4 1 1 1 4 3 2 1
12: 13 1 1 1 1 1 1 1 1 1 1 1 1
13: 14 1 2 1 2 1 2 7 2 1 2 1 2 1
14: 15 1 1 3 1 5 3 1 1 3 5 1 3 1 1
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The rational triangle TinvB(n,m):= A178252(n,m) / T(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10
0: 1
1: 1/2 1
2: 1/3 1 1
3 1/4 1 3/2 1
4: 1/5 1 2 2 1
5: 1/6 1 5/2 10/3 5/2 1
6: 1/7 1 3 5 5 3 1
7: 1/8 1 7/2 7 35/4 7 7/2 1
8: 1/9 1 4 28/3 14 14 28/3 4 1
9: 1/10 1 9/2 12 21 126/5 21 12 9/2 1
10: 1/11 1 5 15 30 42 42 30 15 5 1
Recurrence from the Sheffer a-sequence:
Tinv(3,2) = (3/2)*TinvB(2,1) = (3/2)*1 = 3/2.
From the z-sequence: Tinv(3,0) = 3*Sum_{j=0..2} z_j*TinvB(2,j) = 3*((1/2)*(1/3) -(1/12)*1 + 0*1) = 3*(1/6 - 1/12) = 1/4. - Wolfdieter Lang, Aug 25 2015
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MATHEMATICA
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max = 13; coes = Table[ PadRight[ CoefficientList[ BernoulliB[n, x], x], max], {n, 0, max-1}]; inv = Inverse[coes]; Table[ Take[inv[[n]], n], {n, 1, max}] // Flatten // Denominator (* Jean-François Alcover_, Aug 09 2012 *)
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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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