A130182 -id:A130182 - cp's OEIS frontend

A130185 Third column (m=2) of triangle A130182.

1, 4, 28, 216, -2592, -449280, -42405120, -4187635200, -470931148800, -61395115622400, -9285015785472000, -1621094541631488000, -324607299684433920000, -74032650725692538880000, -19101781491742463754240000, -5540523747380685884620800000

Offset: 0

Wolfdieter Lang Jun 01 2007

See the M. Bruschi et al. reference and the W. Lang link given in A130182.

Cf. A130186 (fourth column).

a(n)= A130185(n+2,2), n>=0.

A130186 Fourth column (m=3) of triangle A130182.

Original entry on oeis.org

1, 20, 508, 17400, 788688, 46032768, 3372731136, 303444195840, 32898989030400, 4229445205094400, 635789661109862400, 110396039984094412800, 21901378996153294848000, 4915686113045739601920000

Offset: 0

Wolfdieter Lang Jun 01 2007

See the M. Bruschi et al. reference and the W. Lang link given in A130182.

Cf. A130185 (third column).

a(n)= A130186(n+3,3), n>=0.

A130183 Row sums of triangle A130182.

Original entry on oeis.org

1, -1, -1, -7, -95, -2105, -68945, -3126895, -187481215, -14354383825, -1366150472225, -158177224078775, -21890843392951775, -3568350888585380425, -676628627780999760625, -147662618352313843681375, -36745377178714846268837375, -10342398978039119692295758625

Offset: 0

Wolfdieter Lang Jun 01 2007

See the Bruschi et al. reference given in A130182.

Cf. A130184 (unsigned row sums).

a(n)=sum(A130182(n,m),m=0..n),n>=0.

A130184 Row sums of unsigned triangle A130182.

Original entry on oeis.org

1, 3, 3, 17, 193, 3655, 109039, 5029265, 303754625, 23283529775, 2209349713375, 254313174846025, 34919610311752225, 5639344047544155575, 1058308036809581999375, 228404316795631157278625

Offset: 0

Wolfdieter Lang Jun 01 2007

See the M. Bruschi et al. reference given in A130182.

Cf. A130183 (row sums).

a(n)=sum(|A130182(n,m)|,m=0..n),n>=0.

A130559 Coefficients of the v=n member of a family of certain orthogonal polynomials with Diophantine properties.

Original entry on oeis.org

1, -2, 1, 12, -8, 1, -144, 108, -20, 1, 2880, -2304, 508, -40, 1, -86400, 72000, -17544, 1708, -70, 1, 3628800, -3110400, 808848, -89280, 4648, -112, 1, -203212800, 177811200, -48405888, 5808528, -349568, 10920, -168, 1, 14631321600, -13005619200, 3663035136, -466619904

Offset: 0

Wolfdieter Lang, Jul 13 2007

For v>=1 the orthogonal polynomials pt(n,v,x) have only integer zeros k*(k+1), k=1..n These integer zeros are from 2*A000217.
Coefficients of pt(n,v=n,x) (in the quoted Bruschi et al. paper {\tilde p}^{(\nu)}_n(x) of eqs. (20) and (24a),(24b)) in increasing powers of x.
The v-family pt(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix Vt=Vt(M,v) with entries Vt_{m,n} given by 2*m*(v+1-m) if n=m, m=1,...,M; -m*(v+1-m) if n=m-1, m=2,...,M; -m*(v+1-m) if n=m+1, m=1..M-1 and 0 else. pt(n,v,x):=det(x*I_n-Vt(n,v) with the n dimensional unit matrix I_n.
pt(n,v=n,x) has, for every n>=1, the n integer zeros 2,6,12,...,n*(n+1). pt(2,2,x) has therefore only the integer zeros 2 and 6. 12= 2*6 = det(Vt(2,2))=16-4.
This triangle coincides with triangle A129467 without row n=0 and column m=0, taking as offset again [0,0].
Column sequences give for m=0..2: A010790(n-1)*(-1)^(n-1), A084915(n+1)*(-1)^n, A130033.

			n=2: [12,-8,1] stands for pt(2,2,x) = 12-8*x+x^2 = (x-2)*(x-6) with the integer zeros 2*1 and 2*3.
Triangle begins:
  [1];
  [-2,1];
  [12,-8,1];
  [-144,108,-20,1];
  [2880,-2304,508,-40,1];
  ...

M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007), pp. 3815-3829.
M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
Wolfdieter Lang, First 10 rows and more.

Row sums give A130031(n+1), n>=0. Unsigned row sums give A130032(n+1), n>=1.

Cf. A130182 (v=1 member).

a(n,m) = [x^m]pt(n,n,x), n>=0, with the three term recurrence for orthogonal polynomial systems of the form pt(n,v,x) = (x + 2*n*(n-1-v))*pt(n-1,v,x) - (n-1)*n*(n-1-v)*(n-2-v)*pt(n-2,v,x), n>=1; pt(-1,v,x) = 0 and pt(0,v,x) = 1. Start with v = n.

cp's OEIS Frontend

A130185 Third column (m=2) of triangle A130182.

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A130186 Fourth column (m=3) of triangle A130182.

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A130183 Row sums of triangle A130182.

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A130184 Row sums of unsigned triangle A130182.

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A130559 Coefficients of the v=n member of a family of certain orthogonal polynomials with Diophantine properties.

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