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 A191302 Denominators in triangle that leads to the Bernoulli numbers. 8
 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 (list; graph; refs; listen; history; text; internal format)
 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 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=1/2; 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 A067132 A207666 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

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Last modified May 27 02:54 EDT 2019. Contains 323597 sequences. (Running on oeis4.)