%I M2178 N0871 #40 Jul 02 2019 14:13:15
%S 1,2,88,3056,319616,18940160,94645408768,526713485312,
%T 2012969145761792,1516106277997969408,950096677725742563328,
%U 125099579935028774699008,1308695886352702185064628224,7547869395875499805522264064
%N From higher order Bernoulli numbers: absolute value of numerator of D Number D2n(2n).
%C 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
%D N. E. Nørlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 462.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Gheorghe Coserea, <a href="/A001904/b001904.txt">Table of n, a(n) for n = 0..200</a>
%H Guodong Liu, <a href="http://dx.doi.org/10.1155/2009/605313">A Recurrence Formula for D Numbers D2n(2n-1)</a>, Discrete Dynamics in Nature and Society, Volume 2009 (2009). [From Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010]
%H N. E. Nørlund, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=237538">Vorlesungen über Differenzenrechnung</a> Springer 1924, p. 462.
%H N. E. Nørlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen über Differenzenrechnung</a> Springer 1924, p. 462.
%H N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924; page 462 [Annotated scanned copy of pages 144-151 and 456-463]
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F 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
%e From Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010: (Start)
%e For n=0 the D-Noerlund number is 1 so a(0)=1.
%e For n=1 the D-Noerlund number is -2/3 so a(1)=2.
%e For n=2 the D-Noerlund number is 88/15 so a(2)=88.
%e For n=3 the D-Noerlund number is -3056/21 so a(3)=3056.
%e For n=4 the D-Noerlund number is 319616/45 so a(4)=319616.
%e For n=5 the D-Noerlund number is -18940160/33 so a(5)=18940160.
%e For n=6 the D-Noerlund number is 94645408768/1365 so a(6)=94645408768, ... .
%e (End)
%p 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
%t NorlundD[nu_, n_] := (-2)^nu NorlundB[nu, n, n/2];
%t Table[NorlundD[2 n, 2 n] // Numerator // Abs, {n, 0, 13}] (* _Jean-François Alcover_, Jul 02 2019 *)
%o (PARI)
%o x='x + O('x^28);
%o abs(apply(numerator, select(i->i, Vec(serlaplace((x / log(x + sqrt(1+x^2))) / sqrt(1+x^2)))))) \\ _Gheorghe Coserea_, Aug 24 2015
%Y Cf. A001905, A261272, A261274 (denominator).
%K nonn,frac
%O 0,2
%A _N. J. A. Sloane_
%E Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 26 2010