A391533 → Pₙ + Pₙ₊₁ → D+1

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Submission text ?

Triangle begins:
  0: 20,  24,    9,     1;
  1: 28,  56,   39,    11,     1;
  2: 36, 104,  113,    57,    13,     1;
  3: 44, 168,  255,   195,    78,    15,     1;
  4: 52, 248,  489,   513,   306,   102,    17,     1;
  5: 60, 344,  839,  1131,   915,   450,   129,    19,    1;
  6: 68, 456, 1329,  2201,  2271,  1503,   631,   159,   21,    1;
  7: 76, 584, 1983,  3907,  4928,  4137,  2324,   853,  192,   23,   1;
  8: 84, 728, 2825,  6465,  9668,  9884,  7014,  3430, 1120,  228,  25,  1;
  9: 92, 888, 3879, 10123, 17541, 21204, 18270, 11250, 4878, 1436, 267, 27, 1;

The row polynomials satisfy (2*n+3+(n+5)*x+x^2)*P(n,x) = (2*n+5+(3*n+11)*x+(n+7)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 20 + 24*x + 9*x^2 + x^3.

P[0, x_] := 20 + 24*x + 9*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 5 + (3*n + 11)*x + (n + 7)*x^2 + x^3)*P[n - 1, x])/(2*n + 3 + (n + 5)*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 20 + 24*x + 9*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 5 + (3*n + 11)*x + (n + 7)*x^2 + x^3)*v[i - 1])/(2*n + 3 + (n + 5)*x + 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 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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.