A127648 → D → D+1

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Submission text ?

Triangle begins:
  0: 0;
  1: 0;
  2: 2;
  3: 6,  6;
  4: 0, 24, 12;
  5: 0,  0, 60,  20;
  6: 0,  0,  0, 120,  30;
  7: 0,  0,  0,   0, 210,  42;
  8: 0,  0,  0,   0,   0, 336,  56;
  9: 0,  0,  0,   0,   0,   0, 504, 72;

The row polynomials satisfy (n^2-5*n+6)*P(n,x) = -(n^2-n)*x^2*P(n-2,x) + (2*n^2-6*n)*x*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2, P(3,x) = 6 + 6*x.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*x^2*P[n - 2, x] + (2*n^2 - 6*n)*x*P[n - 1, x])/(n^2 - 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; v[4] = 6 + 6*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*x^2*v[i - 2] + (2*n^2 - 6*n)*x*v[i - 1])/(n^2 - 5*n + 6)); 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, 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, 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, Laguerre’s theorem on differential operators

Proof chains ? JSON

• Ore-algebra identity of two definitions polytool ? step › direct check
• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the…?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
        • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] 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)
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Veronese sections of Pólya frequency sequences (above)
                • ↑ Parent-row provers of polytool (above)
        • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Veronese sections of Pólya frequency sequences (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
        • ↑ Laguerre’s theorem on differential operators (above)
        • ↑ Laguerre’s theorem on differential operators (above)
• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from one_plus_derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal ma…?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
        • ↑ Laguerre’s theorem on differential operators (above)
        • ↑ Laguerre’s theorem on differential operators (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
        • ↑ Laguerre’s theorem on differential operators (above)
        • ↑ Laguerre’s theorem on differential operators (above)

Related ?

Same fingerprint ?

• A131431 → D+1
• image under θ+1 → (θ+1)² → D+1
• basis-map coefficients of A030528, shifted → D → D+1
• basis-map coefficients of A026729, shifted → (θ+1)² → D+1
• image under θ+1 → θ+1 → D+1 → D
• basis-map coefficients of A021012 → θ+1 → D+1
• basis-map coefficients of A030528, shifted → D+1
• basis-map coefficients of A026729 → (θ+1)² → D+1
• A136572 → D+1
• basis-map coefficients of A026729 → D → D+1
• A130123 → D → D+1
• A131436 → D+1
• basis-map coefficients of A026729, shifted → D → D+1
• A131431 → D+1 → reverse
• A130123 → θ+1 → D+1
• basis-map coefficients of A021012 → D+1
• Hadamard-factor parent of A127773 → D+1
• A127773 → θ+1 → D+1
• A136572 → θ+1 → D+1
• A127773 → D → D+1
• A127648 → (θ+1)² → D+1
• A131431 → D+1 → D
• basis-map coefficients of A026729 → D+1
• A131431 → θ+1 → D+1
• A132440 → θ+1 → D+1
• A136572 → D → D+1
• image under θ+1 → θ+1 → D+1
• basis-map coefficients of A026729, shifted → D+1
• image under θ+1 → D+1
• A130123 → (θ+1)² → D+1
• A131431 → (θ+1)² → D+1
• A130123 → D+1
• basis-map coefficients of A026729, shifted → θ+1 → D+1
• A132440 → D+1
• A127773 → D+1
• basis-map coefficients of A030528, shifted → (θ+1)² → D+1
• basis-map coefficients of A030528, shifted → θ+1 → D+1
• image under θ+1 → θ+1 → D+1 → reverse
• A023531 → D+1
• A136572 → (θ+1)² → D+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.