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 A001904 From higher order Bernoulli numbers: absolute value of numerator of D Number D2n(2n). (Formerly M2178 N0871) 3
 1, 2, 88, 3056, 319616, 18940160, 94645408768, 526713485312, 2012969145761792, 1516106277997969408, 950096677725742563328, 125099579935028774699008, 1308695886352702185064628224, 7547869395875499805522264064 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also, absolute values of reduced numerators of D-Noerlund numbers. By the way, Table 11 from the Nørlund reference (p. 462) gives correctly the first 6 reduced numerators of the D-Noerlund numbers but in the 7th one the author makes a mistake and doesn't divide the numerator (283936226304) and the corresponding denominator (4095) by their common factor (3) to obtain the reduced fraction: 94645408768/1365 which gives the correct value for a(6): 94645408768. - Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010 REFERENCES N. E. Nørlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 462. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Gheorghe Coserea, Table of n, a(n) for n = 0..200 Guodong Liu, A Recurrence Formula for D Numbers D2n(2n-1), Discrete Dynamics in Nature and Society, Volume 2009 (2009). [From Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010] N. E. Nørlund, Vorlesungen über Differenzenrechnung Springer 1924, p. 462. N. E. Nørlund, Vorlesungen über Differenzenrechnung Springer 1924, p. 462. N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 462 [Annotated scanned copy of pages 144-151 and 456-463] FORMULA E.g.f. for D-Noerlund numbers: (x/log(x+sqrt(1+x^2)))/sqrt(1+x^2). - Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010 EXAMPLE From Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010: (Start) For n=0 the D-Noerlund number is 1 so a(0)=1. For n=1 the D-Noerlund number is -2/3 so a(1)=2. For n=2 the D-Noerlund number is 88/15 so a(2)=88. For n=3 the D-Noerlund number is -3056/21 so a(3)=3056. For n=4 the D-Noerlund number is 319616/45 so a(4)=319616. For n=5 the D-Noerlund number is -18940160/33 so a(5)=18940160. For n=6 the D-Noerlund number is 94645408768/1365 so a(6)=94645408768, ... . (End) MAPLE seq(abs(numer(coeff(convert(series((t/log(t+sqrt(1+t^2)))/sqrt(1+t^2), t, 50), polynom), t, 2*n)*(2*n)!)), n=0..23); # Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010 MATHEMATICA NorlundD[nu_, n_] := (-2)^nu NorlundB[nu, n, n/2]; Table[NorlundD[2 n, 2 n] // Numerator // Abs, {n, 0, 13}] (* Jean-François Alcover, Jul 02 2019 *) PROG (PARI) x='x + O('x^28); abs(apply(numerator, select(i->i, Vec(serlaplace((x / log(x + sqrt(1+x^2))) / sqrt(1+x^2)))))) \\ Gheorghe Coserea, Aug 24 2015 CROSSREFS Cf. A001905, A261272, A261274 (denominator). Sequence in context: A058463 A166848 A283631 * A053950 A266182 A012728 Adjacent sequences: A001901 A001902 A001903 * A001905 A001906 A001907 KEYWORD nonn,frac AUTHOR EXTENSIONS Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010 STATUS approved

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Last modified March 23 15:56 EDT 2023. Contains 361445 sequences. (Running on oeis4.)