A026374 → θ+1 → θ+1

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Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1, 12,   9;
  3: 1, 16,  36,   16;
  4: 1, 24,  99,   96,   25;
  5: 1, 28, 153,  272,  175,    36;
  6: 1, 36, 270,  720,  750,   324,   49;
  7: 1, 40, 351, 1200, 1875,  1404,  490,   64;
  8: 1, 48, 522, 2304, 4875,  5184, 2842,  768,   81;
  9: 1, 52, 630, 3232, 8475, 12204, 9898, 4480, 1053, 100;

The row polynomials satisfy P(n,x) = (1+9*x+30*x^2+45*x^3+30*x^4+9*x^5+x^6)*P(n-6,x) - (3+18*x+33*x^2+18*x^3+3*x^4)*P(n-4,x) + (3+9*x+3*x^2)*P(n-2,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 12*x + 9*x^2, P(3,x) = 1 + 16*x + 36*x^2 + 16*x^3, P(4,x) = 1 + 24*x + 99*x^2 + 96*x^3 + 25*x^4, P(5,x) = 1 + 28*x + 153*x^2 + 272*x^3 + 175*x^4 + 36*x^5.

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 12*x + 9*x^2;
P[3, x_] := 1 + 16*x + 36*x^2 + 16*x^3;
P[4, x_] := 1 + 24*x + 99*x^2 + 96*x^3 + 25*x^4;
P[5, x_] := 1 + 28*x + 153*x^2 + 272*x^3 + 175*x^4 + 36*x^5;
P[n_, x_] := P[n, x] = Expand[(1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*P[n - 6, x] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*P[n - 4, x] + (3 + 9*x + 3*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 12*x + 9*x^2; v[4] = 1 + 16*x + 36*x^2 + 16*x^3; v[5] = 1 + 24*x + 99*x^2 + 96*x^3 + 25*x^4; v[6] = 1 + 28*x + 153*x^2 + 272*x^3 + 175*x^4 + 36*x^5; for(n = 6, N, my(i = n + 1); v[i] = (1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*v[i - 6] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*v[i - 4] + (3 + 9*x + 3*x^2)*v[i - 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

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 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) f…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (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.