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%I A178340 #13 Aug 26 2015 05:23:05
%S A178340 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,
%T A178340 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,
%U A178340 1,2,3,4,1,1,1,4,3,2,1,13
%N A178340 Triangle T(n,m) read by rows: denominator of the coefficient [x^m] of the umbral inverse Bernoulli polynomial B^{-1}(n,x).
%C A178340 This is the triangle of denominators associated with the numerators of A178252.
%C A178340 (Unlike the coefficients of the Bernoulli Polynomials, the coefficients of the umbral inverse Bernoulli polynomials are all positive.)
%C A178340 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.
%F A178340 T(n,0) = n+1.
%F A178340 Recurrence for the rational triangle
%F A178340 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
%e A178340 The triangle T(n,m) begins:
%e A178340 n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
%e A178340 0: 1
%e A178340 1: 2 1
%e A178340 2: 3 1 1
%e A178340 3: 4 1 2 1
%e A178340 4: 5 1 1 1 1
%e A178340 5: 6 1 2 3 2 1
%e A178340 6: 7 1 1 1 1 1 1
%e A178340 7: 8 1 2 1 4 1 2 1
%e A178340 8: 9 1 1 3 1 1 3 1 1
%e A178340 9: 10 1 2 1 1 5 1 1 2 1
%e A178340 10: 11 1 1 1 1 1 1 1 1 1 1
%e A178340 11: 12 1 2 3 4 1 1 1 4 3 2 1
%e A178340 12: 13 1 1 1 1 1 1 1 1 1 1 1 1
%e A178340 13: 14 1 2 1 2 1 2 7 2 1 2 1 2 1
%e A178340 14: 15 1 1 3 1 5 3 1 1 3 5 1 3 1 1
%e A178340 ... reformatted. - _Wolfdieter Lang_, Aug 25 2015
%e A178340 -------------------------------------------------
%e A178340 The rational triangle TinvB(n,m):= A178252(n,m) / T(n,m) begins:
%e A178340 n\m 0 1 2 3 4 5 6 7 8 9 10
%e A178340 0: 1
%e A178340 1: 1/2 1
%e A178340 2: 1/3 1 1
%e A178340 3 1/4 1 3/2 1
%e A178340 4: 1/5 1 2 2 1
%e A178340 5: 1/6 1 5/2 10/3 5/2 1
%e A178340 6: 1/7 1 3 5 5 3 1
%e A178340 7: 1/8 1 7/2 7 35/4 7 7/2 1
%e A178340 8: 1/9 1 4 28/3 14 14 28/3 4 1
%e A178340 9: 1/10 1 9/2 12 21 126/5 21 12 9/2 1
%e A178340 10: 1/11 1 5 15 30 42 42 30 15 5 1
%e A178340 ... - _Wolfdieter Lang_, Aug 25 2015
%e A178340 Recurrence from the Sheffer a-sequence:
%e A178340 Tinv(3,2) = (3/2)*TinvB(2,1) = (3/2)*1 = 3/2.
%e A178340 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
%t A178340 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 *)
%Y A178340 Cf. A178252.
%K A178340 nonn,easy,frac,tabl
%O A178340 0,2
%A A178340 _Paul Curtz_, May 25 2010