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A001904
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From higher order Bernoulli numbers: absolute value of numerator of D Number D2n(2n).
(Formerly M2178 N0871)
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3
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1, 2, 88, 3056, 319616, 18940160, 94645408768, 526713485312, 2012969145761792, 1516106277997969408, 950096677725742563328, 125099579935028774699008, 1308695886352702185064628224, 7547869395875499805522264064
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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]
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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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
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EXAMPLE
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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)
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001905, A261272, A261274 (denominator).
Sequence in context: A058463 A166848 A283631 * A053950 A266182 A012728
Adjacent sequences: A001901 A001902 A001903 * A001905 A001906 A001907
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KEYWORD
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nonn,frac
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010
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STATUS
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approved
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