A034870 → Veronese 2 → D

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Submission text ?

Triangle begins:
  0:   0;
  1:   1;
  2:   6,    2;
  3:  15,   30,     3;
  4:  28,  140,    84,      4;
  5:  45,  420,   630,    180,      5;
  6:  66,  990,  2772,   1980,    330,      6;
  7:  91, 2002,  9009,  12012,   5005,    546,     7;
  8: 120, 3640, 24024,  51480,  40040,  10920,   840,    8;
  9: 153, 6120, 55692, 175032, 218790, 111384, 21420, 1224, 9;

The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (2*n^2-n+(6*n^2-11*n)*x)*P(n-1,x) + (4*n*x-4*n*x^2)*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[((2*n^2 - n + (6*n^2 - 11*n)*x)*P[n - 1, x] + (4*n*x - 4*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];
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] = ((2*n^2 - n + (6*n^2 - 11*n)*x)*v[i - 1] + (4*n*x - 4*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3)); 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 veronese_section_2_0 of Even-numbered rows of Pascal's triangle.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from veronese_section_2_0 of Even-numbered rows of Pascal's triangle.?
        • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

          • from Even-numbered rows of Pascal's triangle.?
            • 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
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Even-numbered rows of Pascal's triangle.?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • Geometric basis map (elementary) polytool ? step ›

                  • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                    • 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
                    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                      • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                        • ↑ Parent-row provers of polytool (above)
                        • ↑ Iterated real-rootedness preservers (above)
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • 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
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • ↑ Veronese sections of Pólya frequency sequences (above)

Related ?

Same fingerprint ?

• A091042: Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
• A091042 → reverse
• A103327 → reverse
• A086645 → D
• Hadamard-factor parent of A155495 → D
• A103327: Triangle read by rows: T(n,k) = binomial(2n+1, 2k+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.