A391533 → Pₙ + Pₙ₊₁ → θ+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0: 8,  24,   18,    4;
  1: 8,  40,   54,   28,     5;
  2: 8,  56,  114,  100,    40,     6;
  3: 8,  72,  198,  252,   165,    54,     7;
  4: 8,  88,  306,  516,   480,   252,    70,     8;
  5: 8, 104,  438,  924,  1125,   828,   364,    88,     9;
  6: 8, 120,  594, 1508,  2280,  2178,  1330,   504,   108,   10;
  7: 8, 136,  774, 2300,  4165,  4914,  3871,  2024,   675,  130,   11;
  8: 8, 152,  978, 3332,  7040,  9912,  9604,  6448,  2952,  880,  154,  12;
  9: 8, 168, 1206, 4636, 11205, 18360, 21168, 17424, 10206, 4160, 1122, 180, 13;

The row polynomials satisfy (2+(2*n+4)*x+(n+3)*x^2)*P(n,x) = (2+(2*n+8)*x+(3*n+10)*x^2+(n+4)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 24*x + 18*x^2 + 4*x^3.

P[0, x_] := 8 + 24*x + 18*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (2*n + 8)*x + (3*n + 10)*x^2 + (n + 4)*x^3)*P[n - 1, x])/(2 + (2*n + 4)*x + (n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 8 + 24*x + 18*x^2 + 4*x^3; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (2*n + 8)*x + (3*n + 10)*x^2 + (n + 4)*x^3)*v[i - 1])/(2 + (2*n + 4)*x + (n + 3)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by 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 shift_x_plus_one of Riordan array ((1+x)^3,x).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Riordan array ((1+x)^3,x).?
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array ((1+x)^3,x).?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.