A272866 → Veronese 2 → x ↦ x+1

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Triangle begins:
  0:      1;
  1:      2,       1;
  2:     13,      13,       1;
  3:     62,      93,      33,       1;
  4:    313,     626,     375,      62,       1;
  5:   1562,    3905,    3320,    1075,     100,       1;
  6:   7813,   23439,   25782,   12499,    2490,     147,      1;
  7:  39062,  136717,  184569,  119630,   37513,    4998,    203,     1;
  8: 195313,  781252, 1250002, 1015624,  438475,   95704,   9058,   268,   1;
  9: 976562, 4394529, 8129880, 7946778, 4386294, 1352691, 216132, 15210, 342, 1;

The row polynomials satisfy (2*n-2)*P(n,x) = (5*n^2-7*n+2-(4*n^2-16*n+12)*x)*P(n-1,x) - (20*n-50+(12*n-34)*x-(8*n-18)*x^2)*d/dx P(n-1,x) + (20+40*x+16*x^2-4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 7*n + 2 - (4*n^2 - 16*n + 12)*x)*P[n - 1, x] - (20*n - 50 + (12*n - 34)*x - (8*n - 18)*x^2)*D[P[n - 1, x], x] + (20 + 40*x + 16*x^2 - 4*x^3)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n^2 - 7*n + 2 - (4*n^2 - 16*n + 12)*x)*v[i - 1] - (20*n - 50 + (12*n - 34)*x - (8*n - 18)*x^2)*deriv(v[i - 1], x) + (20 + 40*x + 16*x^2 - 4*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n - 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
            • 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
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
        • ↑ Veronese sections of Pólya frequency sequences (above)
• Ore-algebra identity of two definitions polytool ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
    • ↑ Veronese sections of Pólya frequency sequences (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A272866 → Veronese 2 → x ↦ x/(x+1)

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [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
4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.