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A177427
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Numerators of the Inverse Akiyama-Tanigawa transform of the aerated even-indexed Bernoulli numbers 1, 0, 1/6, 0, -1/30, 0, 1/42,...
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5
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1, 1, 13, 7, 149, 157, 383, 199, 7409, 7633, 86231, 88331, 1173713, 1197473, 1219781, 620401, 42862943, 43503583, 279379879, 283055551, 57313183, 19328341, 449489867, 1362695813, 34409471059, 34738962067, 315510823603, 45467560829, 9307359944587, 9382319148907, 293103346860157, 147643434162641, 594812856101039, 54448301591149
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OFFSET
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0,3
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COMMENTS
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These are the numerators of the first row of a Table T(n,k) which contains the even-indexed Bernoulli numbers in the first column: T(2n,0) = A000367(n)/A002445(n), T(2n+1,0)=0, and which generates rows with the Akiyama-Tanigawa transform. (Because the first column is given, the algorithm is an inverse Akiyama-Tanigawa transform.)
These are the absolute values of the numerators of the Taylor expansion of sinh(ln(x+1))*ln(x+1)at x= 0.[ From Gary Detlefs, Aug 31 2011]
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REFERENCES
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D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05
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LINKS
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Table of n, a(n) for n=0..33.
L. A. Medina, V. H. Moll, E. S. Rowland, Iterated primitives of logarithmic powers, arXiv:0911.1325
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FORMULA
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T(0,k) = H(k)/2+1/(k+1) with H(k) harmonic number of order k. - Roland Groux, Jan 07 2011
T(0,k)= -(1/2)*(k+1)*int(x^n*ln(x*(1-x)),x=0..1). - Roland Groux, Jan 07 2011
G.f.: sum_{k>=0} T(0,k) x^k = (x-2)*(ln(1-x))/(2*x*(1-x)). - Roland Groux, Jan 07 2011
T(1,n) = -A191567(n)/A061038(n+2) = -A060819(n)/A145979(n). - Paul Curtz, Jul 19 2011
(T(1,n))^2 = A181318(n)/A061038(n+2). - Paul Curtz, Jul 19 2011, index corrected by R. J. Mathar, Sep 09 2011
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EXAMPLE
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The table T(n,k) of fractions generated by the Akiyama-Tanigawa transform, with the column T(n,0) equal to bernoulli(n) for even n and equal to 0 for odd n, starts in row n=0 as:
1, 1, 13/12, 7/6, 149/120, 157/120, 383/280, 199/140,...
0, -1/6, -1/4, -3/10, -1/3, -5/14, -3/8, -7/18, -2/5, -9/22,...
1/6, 1/6, 3/20, 2/15, 5/42, 3/28, 7/72, 4/45, 9/110, 5/66,..
0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495, 3/55, 15/286, ...
-1/30, -1/30, -3/140, -1/105, 0, 1/140, 49/3960, 8/495,..
0, -1/42, -1/28, -4/105, -1/28, -29/924, -7/264, -28/1287, -87/5005, ...
1/42, 1/42, 1/140, -1/105, -5/231, -9/308, -343/10296, -1576/45045, ...
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MATHEMATICA
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t[n_, 0] := BernoulliB[n]; t[1, 0]=0; t[n_, k_] := t[n, k] = (t[n, k-1] + (k-1)*t[n, k-1] - t[n+1, k-1])/k; Table[t[0, k], {k, 0, 33}] // Numerator (* Jean-François Alcover, Aug 09 2012 *)
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CROSSREFS
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Cf. A177690 (denominators).
Sequence in context: A217518 A206611 A152142 * A110056 A159562 A076116
Adjacent sequences: A177424 A177425 A177426 * A177428 A177429 A177430
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KEYWORD
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nonn,frac
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AUTHOR
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Paul Curtz, May 07 2010
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STATUS
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approved
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