OFFSET
0,2
LINKS
F. T. Howard, Explicit formulas for degenerate Bernoulli numbers, Discrete Math. 162 (1996), no. 1-3, 175--185. MR1425786 (97m:11024)
FORMULA
The degenerate Bernoulli numbers beta_m(lambda) have e.g.f. x/((1+lambda*x)^(1/lambda)-1).
EXAMPLE
Triangle begins:
1;
-1/2, 1/2;
1/6, 0, -1/6;
0, -1/4, 0, 1/4;
-1/30, 0, 2/3, 0, -19/30;
0, 1/4, 0, -5/2, 0, 9/4;
1/42, 0, -7/4, 0, 12, 0, -863/84;
0, -5/12, 0, 105/8, 0, -70, 0, 1375/24;
...
Thus beta_0(lambda)=1, beta_1(lambda) = -1/2 + lambda/2, ...
PROG
(PARI) cft(n) = {t = x + x*O(x^(n+1)); gf = t/log(1+t); n! * polcoeff(gf, n); } \\ Cauchy numbers first type A006232/A006233
stfk(n, k)=if(n<1, 0, n!*polcoeff(binomial(x, n), k)); \\ Stirling numbers of first kind A008275
polb(m) = if (m==0, 1, if (m==1, -1/2 + 1/2*x, cft(m)*x^m + sum(j=1, m\2, m*bernfrac(2*j)*stfk(m-1, 2*j-1)*x^(m-2*j)/(2*j))));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(denominator(polcoeff(polb(n), k)), ", "); ); ); } \\ Michel Marcus, Feb 16 2014
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Mar 05 2012
EXTENSIONS
More terms from Michel Marcus, Feb 16 2014
STATUS
approved