A064861 → Veronese 2 (j=1) → θ+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:  3;
  1:  5,    8;
  2:  6,   24;
  3:  8,   76,    24;
  4:  9,  126,   108;
  5: 11,  258,   504,    64;
  6: 12,  360,  1080,   384;
  7: 14,  608,  3006,  2368,  160;
  8: 15,  780,  5049,  6240, 1200;
  9: 17, 1180, 10959, 22000, 9200, 384;

The row polynomials satisfy 32*P(n,x) = -(706-2280*x+2208*x^2-256*x^3)*P(n-6,x) + (81-549*x+1044*x^2-576*x^3)*d/dx P(n-6,x) - (888-768*x+384*x^2)*P(n-4,x) - (162+36*x-576*x^2)*d/dx P(n-4,x) + (168+192*x)*P(n-2,x) + (81-144*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 3, P(1,x) = 5 + 8*x, P(2,x) = 6 + 24*x, P(3,x) = 8 + 76*x + 24*x^2, P(4,x) = 9 + 126*x + 108*x^2, P(5,x) = 11 + 258*x + 504*x^2 + 64*x^3.

P[0, x_] := 3;
P[1, x_] := 5 + 8*x;
P[2, x_] := 6 + 24*x;
P[3, x_] := 8 + 76*x + 24*x^2;
P[4, x_] := 9 + 126*x + 108*x^2;
P[5, x_] := 11 + 258*x + 504*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(706 - 2280*x + 2208*x^2 - 256*x^3)*P[n - 6, x] + (81 - 549*x + 1044*x^2 - 576*x^3)*D[P[n - 6, x], x] - (888 - 768*x + 384*x^2)*P[n - 4, x] - (162 + 36*x - 576*x^2)*D[P[n - 4, x], x] + (168 + 192*x)*P[n - 2, x] + (81 - 144*x)*D[P[n - 2, x], x])/(32)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 5 + 8*x; v[3] = 6 + 24*x; v[4] = 8 + 76*x + 24*x^2; v[5] = 9 + 126*x + 108*x^2; v[6] = 11 + 258*x + 504*x^2 + 64*x^3; for(n = 6, N, my(i = n + 1); v[i] = (-(706 - 2280*x + 2208*x^2 - 256*x^3)*v[i - 6] + (81 - 549*x + 1044*x^2 - 576*x^3)*deriv(v[i - 6], x) - (888 - 768*x + 384*x^2)*v[i - 4] - (162 + 36*x - 576*x^2)*deriv(v[i - 4], x) + (168 + 192*x)*v[i - 2] + (81 - 144*x)*deriv(v[i - 2], x))/(32)); 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 veronese_section_2_1 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + …?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.?
        • 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 reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) f…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.?
                • ↑ Iterated real-rootedness preservers (above)
                • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Parent-row provers of polytool (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.