basis-map coefficients of A021012 → Veronese 2 → x ↦ x+1

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Submission text ?

Triangle begins:
   1:     1;
   2:     0;
   3:     2,      2;
   4:     0;
   5:    24,     48,     24;
   6:     0;
   7:   720,   2160,   2160,    720;
   8:     0;
   9: 40320, 161280, 241920, 161280, 40320;
  10:     0;

The row polynomials satisfy P(n,x) = (n^2-3*n+2+(n^2-3*n+2)*x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 0.

P[1, x_] := 1;
P[2, x_] := 0;
P[n_, x_] := P[n, x] = Expand[(n^2 - 3*n + 2 + (n^2 - 3*n + 2)*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 0; for(n = 3, N, my(i = n); v[i] = (n^2 - 3*n + 2 + (n^2 - 3*n + 2)*x)*v[i - 2]); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from veronese_section_2_0 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre pol…?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
            • ↑ Parent-row provers of polytool (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
                • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A136572 → Veronese 2 → x ↦ x+1
• A127648 → Veronese 2 (j=1) → x ↦ x+1
• A136572 → Veronese 2 (j=1) → x ↦ x+1
• A130123 → Veronese 2 (j=1) → x ↦ x+1
• A130123 → Veronese 2 → x ↦ x+1
• A023531 → Veronese 2 → basis A007318
• A127648 → Veronese 2 → 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. [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