A191936 -id:A191936 - 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

A191935 Triangle read by rows of Legendre-Stirling numbers of the second kind.

Original entry on oeis.org

1, 1, 2, 1, 8, 4, 1, 20, 52, 8, 1, 40, 292, 320, 16, 1, 70, 1092, 3824, 1936, 32, 1, 112, 3192, 25664, 47824, 11648, 64, 1, 168, 7896, 121424, 561104, 585536, 69952, 128, 1, 240, 17304, 453056, 4203824, 11807616, 7096384, 419840, 256, 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512

Offset: 1

N. J. A. Sloane, Jun 19 2011

			Triangle begins:
  1;
  1   2;
  1   8    4;
  1  20   52      8;
  1  40  292    320     16;
  1  70 1092   3824   1936     32;
  1 112 3192  25664  47824  11648    64;
  1 168 7896 121424 561104 585536 69952 128;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers
G. E. Andrews et al., The Legendre-Stirling numbers, Discrete Math., 311 (2011), 1255-1272.

Cf. A135921 (row sums), A191936.

Mirror of triangle A071951. - Omar E. Pol, Jan 10 2012

Ps[n_, k_]:= Sum[(-1)^(j+k)*(2*j+1)*j^n*(1+j)^n/((j+k+1)!*(k-j)!), {j,0,k}];
Table[Ps[n, n-k+1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jun 06 2021 *)

T071951(n, k) = sum(i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! );
for (n=1, 10, for (k=1, n, print1(T071951(n,n-k+1), ", ")); print); \\ Michel Marcus, Nov 24 2019

def Ps(n,k): return sum( (-1)^(j+k)*(2*j+1)*j^n*(1+j)^n/(factorial(j+k+1) * factorial(k-j)) for j in (0..k) )
flatten([[Ps(n,n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jun 06 2021

From G. C. Greubel, Jun 06 2021: (Start)
T(n, k) = Ps(n, n-k+1), where Ps(n, k) = Sum_{j=0..k} (-1)^(j+k)*(2*j+1)*j^n*(1 + j)^n/((j+k+1)!*(k-j)!).
Sum_{k=1..n} T(n, k) = A135921(n). (End)

More terms from Omar E. Pol, Jan 10 2012

More terms from Michel Marcus, Nov 24 2019

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

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