A001904 - OEIS

A001904

From higher order Bernoulli numbers: absolute value of numerator of D Number D2n(2n).
(Formerly M2178 N0871)

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. 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.

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: A364974 A166848 A283631 * A053950 A266182 A362519

Adjacent sequences: A001901 A001902 A001903 * A001905 A001906 A001907

KEYWORD