A106246 → D+1 → (θ+1)²

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Submission text ?

Triangle begins:
  0: 1;
  1: 3,  4;
  2: 5, 24,   9;
  3: 3, 60,  81,  16;
  4: 0, 48, 270, 192,   25;
  5: 0,  0, 270, 800,  375,   36;
  6: 0,  0,   0, 960, 1875,  648,    49;
  7: 0,  0,   0,   0, 2625, 3780,  1029,    64;
  8: 0,  0,   0,   0,    0, 6048,  6860,  1536,   81;
  9: 0,  0,   0,   0,    0,    0, 12348, 11520, 2187, 100;

The row polynomials satisfy P(n,x) = -x^6*P(n-6,x) + 6*x^5*P(n-5,x) - 15*x^4*P(n-4,x) + 20*x^3*P(n-3,x) - 15*x^2*P(n-2,x) + 6*x*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 5 + 24*x + 9*x^2, P(3,x) = 3 + 60*x + 81*x^2 + 16*x^3, P(4,x) = 48*x + 270*x^2 + 192*x^3 + 25*x^4, P(5,x) = 270*x^2 + 800*x^3 + 375*x^4 + 36*x^5.

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 5 + 24*x + 9*x^2;
P[3, x_] := 3 + 60*x + 81*x^2 + 16*x^3;
P[4, x_] := 48*x + 270*x^2 + 192*x^3 + 25*x^4;
P[5, x_] := 270*x^2 + 800*x^3 + 375*x^4 + 36*x^5;
P[n_, x_] := P[n, x] = Expand[-x^6*P[n - 6, x] + 6*x^5*P[n - 5, x] - 15*x^4*P[n - 4, x] + 20*x^3*P[n - 3, x] - 15*x^2*P[n - 2, x] + 6*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; v[3] = 5 + 24*x + 9*x^2; v[4] = 3 + 60*x + 81*x^2 + 16*x^3; v[5] = 48*x + 270*x^2 + 192*x^3 + 25*x^4; v[6] = 270*x^2 + 800*x^3 + 375*x^4 + 36*x^5; for(n = 6, N, my(i = n + 1); v[i] = -x^6*v[i - 6] + 6*x^5*v[i - 5] - 15*x^4*v[i - 4] + 20*x^3*v[i - 3] - 15*x^2*v[i - 2] + 6*x*v[i - 1]); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from one_plus_derivative of Number triangle T(n,k)=C(n,k)C(2,n-k).?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Number triangle T(n,k)=C(n,k)C(2,n-k).?
        • 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)
        • 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
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Number triangle T(n,k)=C(n,k)C(2,n-k).?
        • ↑ Laguerre’s theorem on differential operators (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

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.