A102537 → θ+1

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Triangle begins:
   1: 1;
   2: 1,   6;
   3: 1,  16,   36;
   4: 1,  30,  165,    220;
   5: 1,  48,  468,   1456,    1365;
   6: 1,  70, 1050,   5600,   11900,    8568;
   7: 1,  96, 2040,  16320,   58140,   93024,    54264;
   8: 1, 126, 3591,  39900,  209475,  553014,   706629,   346104;
   9: 1, 160, 5880,  86240,  619850, 2380224,  4958800,  5262400,  2220075;
  10: 1, 198, 9108, 170016, 1593900, 8288280, 24864840, 42625440, 38629305, 14307150;

The row polynomials satisfy (4*n^3-2*n^2-2*n)*P(n,x) = (4*n^3-2*n^2-2*n+(18*n^3-6*n^2)*x)*P(n-1,x) + ((8*n^2-n-1)*x+(9*n^2-3*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 2*n^2 - 2*n + (18*n^3 - 6*n^2)*x)*P[n - 1, x] + ((8*n^2 - n - 1)*x + (9*n^2 - 3*n)*x^2)*D[P[n - 1, x], x])/(4*n^3 - 2*n^2 - 2*n)]];
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] = ((4*n^3 - 2*n^2 - 2*n + (18*n^3 - 6*n^2)*x)*v[i - 1] + ((8*n^2 - n - 1)*x + (9*n^2 - 3*n)*x^2)*deriv(v[i - 1], x))/(4*n^3 - 2*n^2 - 2*n)); 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 Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.?
            • 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
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
            • 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
    • ↑ Iterated real-rootedness preservers (above)
    • ↑ Row-hypergeometric rows (multiplier sequences) (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.