A131689 → θ+1 → (θ+1)²

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
   1: 1;
   2: 0, 8;
   3: 0, 8,    54;
   4: 0, 8,   162,     384;
   5: 0, 8,   378,    2304,     3000;
   6: 0, 8,   810,    9600,    30000,     25920;
   7: 0, 8,  1674,   34560,   195000,    388800,    246960;
   8: 0, 8,  3402,  115584,  1050000,   3628800,   5186160,    2580480;
   9: 0, 8,  6858,  370944,  5103000,  27216000,  65691360,   72253440,   29393280;
  10: 0, 8, 13770, 1161600, 23310000, 180169920, 653456160, 1192181760, 1058158080, 362880000;

The row polynomials satisfy P(n,x) = 22*x^2*P(n-2,x) - (4*x^2-26*x^3)*d/dx P(n-2,x) - (x^2-4*x^3-5*x^4)*d^2/dx^2 P(n-2,x) - (1-8*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 8*x.

P[1, x_] := 1;
P[2, x_] := 8*x;
P[n_, x_] := P[n, x] = Expand[22*x^2*P[n - 2, x] - (4*x^2 - 26*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3 - 5*x^4)*D[P[n - 2, x], {x, 2}] - (1 - 8*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 8*x; for(n = 3, N, my(i = n); v[i] = 22*x^2*v[i - 2] - (4*x^2 - 26*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3 - 5*x^4)*deriv(deriv(v[i - 2], x), x) - (1 - 8*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from theta_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 …?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.?
        • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.?
        • ↑ Ma–Wang interval theorem (above)
        • ↑ Iterated real-rootedness preservers (above)

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.