A192456 Numerators in triangle that leads to the Bernoulli numbers.

1, 1, 1, -1, 1, -1, 1, -2, 2, 1, -5, 1, 1, -1, 3, -8, 1, -7, 14, -4, 1, -4, 4, -64, 8, 1, -3, 9, -8, 12, 1, -5, 7, -40, 20, -32, 1, -11, 44, -44, 44, -16, 1, -2, 18, -64, 4, -192, 6112

Offset

0,8

Comments

For the denominators and detailed information see A191302.

Links

Table of n, a(n) for n=0..48.

Maple

nmax:=14: mmax:=nmax: A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end: A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end: for m from 0 to 2*mmax do T(0, m):=A164555(m)/A027642(m) od: for n from 1 to nmax do for m from 0 to 2*mmax do T(n, m):=T(n-1, m+1)-T(n-1, m) od: od: seq(T(n, n+1), n=0..nmax): for n from 0 to nmax do ASPEC(n, 0):=2: for m from 1 to mmax do ASPEC(n, m):= (2*n+m)*binomial(n+m-1, m-1)/m od: od: for n from 0 to nmax do seq(ASPEC(n, m), m=0..mmax) od: for n from 0 to nmax do for m from 0 to 2*mmax do SBD(n, m):=0 od: od: for m from 0 to mmax do for n from 2*m to nmax do SBD(n, m):= T(m, m+1) od: od: for n from 0 to nmax do seq(SBD(n, m), m= 0..mmax/2) od: for n from 0 to nmax do BSPEC(n, 2) := SBD(n, 2)*ASPEC(2, n-4) od: for m from 0 to mmax do for n from 0 to nmax do BSPEC(n, m) := SBD(n, m)*ASPEC(m, n-2*m) od: od: for n from 0 to nmax do seq(BSPEC(n, m), m=0..mmax/2) od: seq(add(BSPEC(n, k), k=0..floor(n/2)) , n=0..nmax): Tx:=0: for n from 0 to nmax do for m from 0 to floor(n/2) do a(Tx):= numer(BSPEC(n, m)): Tx:=Tx+1: od: od: seq(a(n), n=0..Tx-1); # Johannes W. Meijer, Jul 02 2011

Mathematica

(* a=ASPEC, b=BSPEC *) nmax = 13; a[n_, 0] = 2; a[n_, m_] := (2n+m)*Binomial[n+m-1, m-1]/m; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, nmax}]; diff = Table[ Differences[bb, n], {n, 1, nmax}]; dd = Diagonal[diff]; sbd[n_, m_] := If[n >= 2m, -dd[[m+1]], 0]; b[n_, m_] := sbd[n, m]*a[m, n-2m]; Table[b[n, m], {n, 0, nmax}, {m, 0, Floor[n/2]}] // Flatten // Numerator (* Jean-François Alcover, Aug 09 2012 *)

Crossrefs

Cf. A191302 (denominators).

Sequence in context: A291260 A350825 A218529 * A226948 A010243 A332963

Adjacent sequences: A192453 A192454 A192455 * A192457 A192458 A192459

Keyword

sign,frac,tabf

Author

Paul Curtz, Jul 01 2011

Extensions

Edited and Maple program added by Johannes W. Meijer, Jul 02 2011

Status

approved