A191662 -id:A191662 - cp's OEIS frontend

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

Paul Curtz, May 30 2011

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.

			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

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

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

(* 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 *)

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).

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

A052612 Expansion of e.g.f. x*(2+x)/(1-x^2).

Original entry on oeis.org

0, 2, 2, 12, 24, 240, 720, 10080, 40320, 725760, 3628800, 79833600, 479001600, 12454041600, 87178291200, 2615348736000, 20922789888000, 711374856192000, 6402373705728000, 243290200817664000, 2432902008176640000, 102181884343418880000, 1124000727777607680000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Stirling transform of (-1)^n*a(n-1) = [0,2,-2,12,-24,...] is A052856(n-1) =[0,2,4,14,76,...]. - Michael Somos, Mar 04 2004

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 557.

Cf. A000034, A000142, A191662.

spec := [S,{S=Prod(Z,Union(Sequence(Z),Sequence(Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[x (2+x)/(1-x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 10 2018 *)
Join[{0}, Table[1/2 (3 - (-1)^n) n!, {n, 20}]] (* David Trimas, Jul 28 2023 *)

a(n)=if(n<0,0,n!*polcoeff((x^2+2*x)/(1-x^2)+x*O(x^n),n))

PARI
```
a(n)=if(n<1,0,n!*(n%2+1))
```

a(n)= n! / gcd(n, n * (n + 1) / 2) \\ Andrew S. Plewe, Jan 09 2006

Recurrence: {a(0)=0, a(1)=2, a(2)=2, (-2-n^2-3*n)*a(n)+a(n+2)=0}.
Sum(1/2*(2+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2))*n!.
E.g.f.: x*(x+2)/(1-x^2).
a(2n+1) = 2*(2n+1)!, a(2n) = (2n)!, if n>0.
a(n) = n! if n is even, 2*n! otherwise. a(n) = n!*A000034(n).
a(n) = n! / gcd(n, T(n)) where T(n) is the n-th triangular number. - Andrew S. Plewe, Jan 09 2006
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = sinh(1)/2 + cosh(1) - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = sinh(1)/2 - cosh(1) + 1. (End)
a(0)=0, a(n) = (1/2)*(3 - (-1)^n)*n! if n>0. - David Trimas, Jul 28 2023
a(n) = 2 * A191662(n) for n>=1. - Alois P. Heinz, Sep 05 2023

a(20)-a(22) from Alois P. Heinz, Sep 05 2023

A192575 Triangle T(n,0) = A040000(n), T(n,k)=0 (odd-numbered columns); T(n,k) = (-1)^(k/2)*A110813(n-k/2-1,k/2-1) (even-numbered columns, k>0).

Original entry on oeis.org

1, 2, 0, 2, 0, -1, 2, 0, -3, 0, 2, 0, -5, 0, 1, 2, 0, -7, 0, 4, 0, 2, 0, -9, 0, 9, 0, -1, 2, 0, -11, 0, 16, 0, -5, 0, 2, 0, -13, 0, 25, 0, -14, 0, 1, 2, 0, -15, 0, 36, 0, -30, 0, 6, 0, 2, 0, -17, 0, 49, 0, -55, 0, 20, 0, -1

Offset: 0

Paul Curtz, Jul 04 2011

A zero-padded variant of A110813, which provides more information.

			1;
2 0;
2 0  -1;
2 0  -3 0;
2 0  -5 0  1;
2 0  -7 0  4 0;
2 0  -9 0  9 0  -1;
2 0 -11 0 16 0  -5 0;
2 0 -13 0 25 0 -14 0 1;
2 0 -15 0 36 0 -30 0 6 0;

Cf. A191662.

T(n,k) = T(n-1,k)-T(n-2,k-2), n>1.
T(n,2k+1)=0.
T(n,2k) = (-1)^k*binomial(n-k-1,k-1)*(2n-3k)/k , k>0. - R. J. Mathar, Aug 26 2011
T(n,0) = A040000(n).
sum_{k=0..n} T(n,k) = A057079(n).
sum_{k=0..n} |T(n,k)| = A000045(n+2). (See A129710).

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A191302 Denominators in triangle that leads to the Bernoulli numbers.

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A052612 Expansion of e.g.f. x*(2+x)/(1-x^2).

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A192575 Triangle T(n,0) = A040000(n), T(n,k)=0 (odd-numbered columns); T(n,k) = (-1)^(k/2)*A110813(n-k/2-1,k/2-1) (even-numbered columns, k>0).

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