A130033 -id:A130033 - cp's OEIS frontend

A129467 Orthogonal polynomials with all zeros integers from 2*A000217.

1, 0, 1, 0, -2, 1, 0, 12, -8, 1, 0, -144, 108, -20, 1, 0, 2880, -2304, 508, -40, 1, 0, -86400, 72000, -17544, 1708, -70, 1, 0, 3628800, -3110400, 808848, -89280, 4648, -112, 1, 0, -203212800, 177811200, -48405888, 5808528, -349568, 10920, -168, 1, 0, 14631321600, -13005619200, 3663035136, -466619904, 30977424, -1135808, 23016, -240, 1

Offset: 0

Wolfdieter Lang, May 04 2007

The row polynomials p(n,x) = Sum_{k=0..n} T(n,k)*x^k have the n integer zeros 2*A000217(j), j=0..n-1.
The row polynomials satisfy a three-term recurrence relation which qualify them as orthogonal polynomials w.r.t. some (as yet unknown) positive measure.
Column sequences (without leading zeros) give A000007, A010790(n-1)*(-1)^(n-1), A084915(n-1)*(-1)^(n-2), A130033 for m=0..3.
Apparently this is the triangle read by rows of Legendre-Stirling numbers of the first kind. See the Andrews-Gawronski-Littlejohn paper, table 2. The mirror version is the triangle A191936. - Omar E. Pol, Jan 10 2012

			Triangle starts:
  1;
  0,    1;
  0,   -2,     1;
  0,   12,    -8,   1;
  0, -144,   108, -20,   1;
  0, 2880, -2304, 508, -40,  1;
  ...
n=3: [0,12,-8,1]. p(3,x) = x*(12-8*x+x^2) = x*(x-2)*(x-6).
n=5: [0,2880,-2304,508,-40,1]. p(5,x) = x*(2880-2304*x+508*x^2-40*x^3 +x^4) = x*(x-2)*(x-6)*(x-12)*(x-20).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
G. E. Andrews, W. Gawronski, and L. L. Littlejohn, The Legendre-Stirling Numbers, Discrete Mathematics, Volume 311, Issue 14, 28 July 2011, Pages 1255-1272.
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.
José L. Cereceda, A refinement of Lang's formula for the sum of powers of integers, arXiv:2301.02141 [math.NT], 2023.
José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 14.
Mark W. Coffey and Matthew 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 ten rows and more.

Cf. A129462 (v=2 member), A129065 (v=1 member), A191936 (row reversed?).
Cf. A000217, A130031 (row sums), A130032 (unsigned row sums), A191936.
Column sequences (without leading zeros): A000007 (k=0), (-1)^(n-1)*A010790(n-1) (k=1), (-1)^n*A084915(n-1) (k=2), A130033 (k=3).
Cf. A008275.

f:= func< n,k | (&+[Binomial(2*k+j,j)*StirlingFirst(2*n,2*k+j)*n^j: j in [0..2*(n-k)]]) >;
function T(n,k) // T = A129467
  if k eq n then return 1;
  else return f(n,k) -  (&+[Binomial(j,2*(j-k))*T(n,j): j in [k+1..n]]);
end if;
end function;
[[T(n,k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 09 2024

T[n_, k_, m_]:= T[n,k,m]= If[k<0 || k>n, 0, If[n==0, 1, (2*(n-1)*(n-m) -(m-1))*T[n-1,k,m] -((n-1)*(n-m-1))^2*T[n-2,k,m] +T[n-1,k-1,m]]]; (* T=A129467 *)
Table[T[n,k,n], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 09 2024 *)

@CachedFunction
def f(n,k): return sum(binomial(2*k+j,j)*(-1)^j*stirling_number1(2*n,2*k+j)*n^j for j in range(2*n-2*k+1))
def T(n,k): # T = A129467
    if n==0: return 1
    else: return - sum(binomial(j,2*j-2*k)*T(n,j) for j in range(k+1,n+1)) + f(n,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 09 2024

Row polynomials p(n,x) = Product_{m=1..n} (x - m*(m-1)), n>=1, with p(0,x) = 1.
Row polynomials p(n,x) = p(n, v=n, x) with the recurrence: p(n,v,x) = (x + 2*(n-1)^2 - 2*(v-1)*(n-1) - v + 1)*p(n-1,v,x) - (n-1)^2*(n-1-v)^2*p(n-2,v,x) with p(-1,v,x) = 0 and p(0,v,x) = 1.
T(n, k) = [x^k] p(n, n, x), n >= k >= 0, otherwise 0.
T(n, k) = Sum_{j=0..2*(n-k)} ( binomial(2*k+j, j)*s(n,k)*n^j ) - Sum_{j=k+1..n} binomial(j, 2*(j-k))*T(n, j) (See Coffey and Lettington formula (4.7)). - G. C. Greubel, Feb 09 2024

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

A129467 Orthogonal polynomials with all zeros integers from 2*A000217.

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