Veronese parent of A098172 → θ+1

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Submission text ?

Triangle begins:
   1: 1;
   2: 0, 0, 3, 4;
   3: 0, 0, 0, 0, 5, 12, 7;
   4: 0, 0, 0, 0, 0,  0, 7, 24, 27, 10;
   5: 0, 0, 0, 0, 0,  0, 0,  0,  9, 40, 66, 48,  13;
   6: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 11, 60, 130, 140,  75,  16;
   7: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  13,  84, 225, 320, 255, 108,  19;
   8: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  15, 112, 357, 630, 665,  420,  147,   22;
   9: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  17, 144, 532, 1120, 1470, 1232,  644,  192,   25;
  10: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  19,  180,  756, 1848, 2898, 3024, 2100, 936, 243, 28;

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

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

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((2*n - 1)*x^2 + (5*n - 3)*x^3 + (3*n - 2)*x^4)*v[i - 1])/(2*n - 3 + (3*n - 5)*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, Laguerre’s theorem on differential operators
• 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, Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences, Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • 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
• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
    • 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
    • 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

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.