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 A178340 Triangle T(n,m) read by rows: denominator of the coefficient [x^m] of the umbral inverse Bernoulli polynomial B^{-1}(n,x). 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS FORMULA 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 EXAMPLE 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 ... reformatted. - Wolfdieter Lang, Aug 25 2015 ------------------------------------------------- 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 ... - Wolfdieter Lang, Aug 25 2015 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 MATHEMATICA 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 *) CROSSREFS Cf. A178252. Sequence in context: A263646 A113924 A262891 * A173261 A084296 A209235 Adjacent sequences:  A178337 A178338 A178339 * A178341 A178342 A178343 KEYWORD nonn,easy,frac,tabl AUTHOR Paul Curtz, May 25 2010 STATUS approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)