This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A191302 #17 Feb 24 2019 21:09:37
%S A191302 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,
%T A191302 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,
%U A191302 33,1155
%N A191302 Denominators in triangle that leads to the Bernoulli numbers.
%C A191302 For the definition of the ASPEC array coefficients see the formulas; see also A029635 (Lucas triangle), A097207 and A191662 (k-dimensional square pyramidal numbers).
%C A191302 The antidiagonal row sums of the ASPEC array equal A042950(n) and A098011(n+3).
%C A191302 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.
%C A191302 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.
%C A191302 The row sums of the BSPEC triangle give the Bernoulli numbers A164555(n)/A027642(n).
%C A191302 For the numerators of the BSPEC triangle coefficients see A192456.
%F A191302 ASPEC(n, 0) = 2 and ASPEC(n, m) = (2*n+m)*binomial(n+m-1, m-1)/m, n >= 0, m >= 1.
%F A191302 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.
%F A191302 SBD(n, m) = T(m, m+1), n >= 2*m; see A190339 for the definition of the T(n, m).
%F A191302 BSPEC(n, m) = SBD(n, m)*ASPEC(m, n-2*m)
%F A191302 Sum_{k=0..floor(n/2)} BSPEC(n, k) = A164555(n)/A027642(n).
%e A191302 The first few rows of the array ASPEC array:
%e A191302 2, 1, 1, 1, 1, 1, 1,
%e A191302 2, 3, 4, 5, 6, 7, 8,
%e A191302 2, 5, 9, 14, 20, 27, 35,
%e A191302 2, 7, 16, 30, 50, 77, 112,
%e A191302 2, 9, 25, 55, 105, 182, 294,
%e A191302 The first few T(n,n+1) = T(n,n)/2 coefficients:
%e A191302 1/2, -1/6, 1/15, -4/105, 4/105, -16/231, 3056/15015, ...
%e A191302 The first few rows of the SBD array:
%e A191302 1/2, 0, 0, 0
%e A191302 1/2, 0, 0, 0
%e A191302 1/2, -1/6, 0, 0
%e A191302 1/2, -1/6, 0, 0
%e A191302 1/2, -1/6, 1/15, 0
%e A191302 1/2, -1/6, 1/15, 0
%e A191302 1/2, -1/6, 1/15, -4/105
%e A191302 1/2, -1/6, 1/15, -4/105
%e A191302 The first few rows of the BSPEC triangle:
%e A191302 B(0) = 1 = 1/1
%e A191302 B(1) = 1/2 = 1/2
%e A191302 B(2) = 1/6 = 1/2 - 1/3
%e A191302 B(3) = 0 = 1/2 - 1/2
%e A191302 B(4) = -1/30 = 1/2 - 2/3 + 2/15
%e A191302 B(5) = 0 = 1/2 - 5/6 + 1/3
%e A191302 B(6) = 1/42 = 1/2 - 1/1 + 3/5 - 8/105
%e A191302 B(7) = 0 = 1/2 - 7/6 + 14/15 - 4/15
%p A191302 nmax:=13: mmax:=nmax:
%p A191302 A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end:
%p A191302 A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end:
%p A191302 for m from 0 to 2*mmax do T(0,m):=A164555(m)/A027642(m) od:
%p A191302 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:
%p A191302 seq(T(n,n+1),n=0..nmax):
%p A191302 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:
%p A191302 for n from 0 to nmax do seq(ASPEC(n,m),m=0..mmax) od:
%p A191302 for n from 0 to nmax do for m from 0 to 2*mmax do SBD(n,m):=0 od: od:
%p A191302 for m from 0 to mmax do for n from 2*m to nmax do SBD(n,m):= T(m,m+1) od: od:
%p A191302 for n from 0 to nmax do seq(SBD(n,m), m= 0..mmax/2) od:
%p A191302 for n from 0 to nmax do BSPEC(n,2) := SBD(n,2)*ASPEC(2,n-4) od:
%p A191302 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:
%p A191302 for n from 0 to nmax do seq(BSPEC(n,m), m=0..mmax/2) od:
%p A191302 seq(add(BSPEC(n, k), k=0..floor(n/2)) ,n=0..nmax):
%p A191302 Tx:=0:
%p A191302 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:
%p A191302 seq(a(n),n=0..Tx-1); # _Johannes W. Meijer_, Jul 02 2011
%t A191302 (* 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 *)
%Y A191302 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
%K A191302 nonn,frac,tabf
%O A191302 0,2
%A A191302 _Paul Curtz_, May 30 2011
%E A191302 Edited, Maple program and crossrefs added by _Johannes W. Meijer_, Jul 02 2011