A064861 → reverse → x ↦ x+1

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Submission text ?

Triangle begins:
  0:     6,      5,      1;
  1:    18,     21,      8,      1;
  2:    36,     60,     37,     10,      1;
  3:   108,    216,    171,     67,     13,      1;
  4:   216,    540,    558,    305,     93,     15,     1;
  5:   648,   1836,   2214,   1473,    584,    138,    18,     1;
  6:  1296,   4320,   6264,   5160,   2641,    860,   174,    20,    1;
  7:  3888,  14256,  23112,  21744,  13083,   5221,  1382,   234,   23,   1;
  8:  7776,  32400,  60480,  66600,  47910,  23525,  7985,  1850,  280,  25,  1;
  9: 23328, 104976, 213840, 260280, 210330, 118485, 47480, 13535, 2690, 355, 28, 1;

The row polynomials satisfy P(n,x) = -(5*n+2+(2*n+1)*x)*P(n-1,x) + (12+10*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 5*x + x^2.

P[0, x_] := 6 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(5*n + 2 + (2*n + 1)*x)*P[n - 1, x] + (12 + 10*x + 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(5*n + 2 + (2*n + 1)*x)*v[i - 1] + (12 + 10*x + 2*x^2)*deriv(v[i - 1], x)); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 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 [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…?
            no recorded property covers this requirement
        • Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

• A103633 → basis A013609 → x ↦ x/(x+1)
• A152842 → basis A013611
• A064861 → x ↦ x/(x+1)

References

1. [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
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.