A191302 Denominators in triangle that leads to the Bernoulli numbers.

1, 2, 2, 3, 2, 2, 2, 3, 15, 2, 6, 3, 2, 1, 5, 105, 2, 6, 15, 15, 2, 3, 3, 105, 105, 2, 2, 5, 7, 35, 2, 3, 3, 21, 21, 231, 2, 6, 15, 15, 21, 21, 2, 1, 5, 15, 1, 77, 15015, 2, 6, 3, 35, 15, 33, 1155

Offset

0,2

Comments

For the definition of the ASPEC array coefficients see the formulas; see also A029635 (Lucas triangle), A097207 and A191662 (k-dimensional square pyramidal numbers).

The antidiagonal row sums of the ASPEC array equal A042950(n) and A098011(n+3).

The coefficients of the T(n,m) array are defined in A190339. We define the coefficients of the SBD array with the aid of the T(n,n+1), see the formulas and the examples.

Multiplication of the coefficients in the rows of the ASPEC array with the coefficients in the columns of the SBD array leads to the coefficients of the BSPEC triangle, see the formulas. The BSPEC triangle can be looked upon as a spectrum for the Bernoulli numbers.

The row sums of the BSPEC triangle give the Bernoulli numbers A164555(n)/A027642(n).

For the numerators of the BSPEC triangle coefficients see A192456.

Links

Table of n, a(n) for n=0..55.

Formula

ASPEC(n, 0) = 2 and ASPEC(n, m) = (2*n+m)*binomial(n+m-1, m-1)/m, n >= 0, m >= 1.

ASPEC(n, m) = ASPEC(n-1, m) + ASPEC(n, m-1), n >= 1, m >= 1, with ASPEC(n, 0) = 2, n >= 0, and ASPEC(0,m) = 1, m >= 1.

SBD(n, m) = T(m, m+1), n >= 2*m; see A190339 for the definition of the T(n, m).

BSPEC(n, m) = SBD(n, m)*ASPEC(m, n-2*m)

Sum_{k=0..floor(n/2)} BSPEC(n, k) = A164555(n)/A027642(n).

Example

The first few rows of the array ASPEC array:
  2, 1,  1,  1,   1,   1,    1,
  2, 3,  4,  5,   6,   7,    8,
  2, 5,  9, 14,  20,  27,   35,
  2, 7, 16, 30,  50,  77,  112,
  2, 9, 25, 55, 105, 182,  294,
The first few T(n,n+1) = T(n,n)/2 coefficients:
1/2, -1/6, 1/15, -4/105, 4/105, -16/231, 3056/15015, ...
The first few rows of the SBD array:
  1/2,   0,   0,     0
  1/2,   0,   0,     0
  1/2, -1/6,  0,     0
  1/2, -1/6,  0,     0
  1/2, -1/6, 1/15,   0
  1/2, -1/6, 1/15,   0
  1/2, -1/6, 1/15, -4/105
  1/2, -1/6, 1/15, -4/105
The first few rows of the BSPEC triangle:
  B(0) =   1   = 1/1
  B(1) =  1/2  = 1/2
  B(2) =  1/6  = 1/2 - 1/3
  B(3) =   0   = 1/2 - 1/2
  B(4) = -1/30 = 1/2 - 2/3 +  2/15
  B(5) =   0   = 1/2 - 5/6 +  1/3
  B(6) =  1/42 = 1/2 - 1/1 +  3/5  - 8/105
  B(7) =   0   = 1/2 - 7/6 + 14/15 - 4/15

Maple

nmax:=13: 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):= denom(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 // Denominator (* Jean-François Alcover_, Aug 09 2012 *)

Crossrefs

Cf. A028246 (Worpitzky), A085737/A085738 (Conway-Sloane) and A051714/A051715 (Akiyama-Tanigawa) for other triangles that lead to the Bernoulli numbers. - Johannes W. Meijer, Jul 02 2011

Sequence in context: A060244 A072814 A196229 * A161189 A328162 A375767

Adjacent sequences: A191299 A191300 A191301 * A191303 A191304 A191305

Keyword

nonn,frac,tabf

Author

Paul Curtz, May 30 2011

Extensions

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

Status

approved