A209123 Triangle read by rows: numerators of degenerate Bernoulli numbers written as powers of lambda.

1, -1, 1, 1, 0, -1, 0, -1, 0, 1, -1, 0, 2, 0, -19, 0, 1, 0, -5, 0, 9, 1, 0, -7, 0, 12, 0, -863, 0, -5, 0, 105, 0, -70, 0, 1375, -1, 0, 50, 0, -1624, 0, 480, 0, -33953, 0, 21, 0, -70, 0, 9849, 0, -3780, 0, 57281, 5, 0, -91, 0, 5345, 0, -29531, 0, 33600, 0, -3250433

Offset

0,13

Comments

The constant terms are the regular Bernoulli numbers A027641/A027642.

Links

Table of n, a(n) for n=0..65.

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 β_m(λ) have e.g.f. x/((1+λ*x)^(1/λ)-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 β_0(λ)=1, β_1(λ) = -1/2 + λ/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(numerator(polcoeff(polb(n), k)), ", "); ); ); } \\ Michel Marcus, Feb 16 2014

Crossrefs

Cf. A209124, A027641/A027642.

Sequence in context: A361591 A261516 A009198 * A139003 A264881 A280622

Adjacent sequences: A209120 A209121 A209122 * A209124 A209125 A209126

Keyword

sign,frac,tabl

Author

N. J. A. Sloane, Mar 05 2012

Extensions

More terms from Michel Marcus, Feb 16 2014

Status

approved