A108426 → D

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Triangle begins:
  0:       0;
  1:       1;
  2:       5,       4;
  3:      28,      42,       15;
  4:     165,     360,      252,       56;
  5:    1001,    2860,     2970,     1320,      210;
  6:    6188,   21840,    30030,    20020,     6435,      792;
  7:   38760,  162792,   278460,   247520,   120120,    30030,    3003;
  8:  245157, 1193808,  2441880,  2713200,  1763580,   668304,  136136,  11440;
  9: 1562275, 8652600, 20593188, 27457584, 22383900, 11395440, 3527160, 604656, 43758;

The row polynomials satisfy (4*n^3-6*n^2+2*n)*P(n,x) = (27*n^3-54*n^2+33*n-6+(24*n^3-52*n^2+36*n-8)*x+(n^3-4*n^2+5*n-2)*x^2)*P(n-1,x) - ((9*n^2-9*n+2)*x+(10*n^2-11*n+3)*x^2+(n^2-2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 52*n^2 + 36*n - 8)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*P[n - 1, x] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 11*n + 3)*x^2 + (n^2 - 2*n + 1)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 + 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 52*n^2 + 36*n - 8)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*v[i - 1] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 11*n + 3)*x^2 + (n^2 - 2*n + 1)*x^3)*deriv(v[i - 1], x))/(4*n^3 - 6*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 - 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
    • 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, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • 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.