A064861 → x ↦ x+1 → reverse

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Triangle begins:
  0:  2,    7,     6;
  1:  4,   20,    33,    18;
  2:  4,   28,    73,    84,     36;
  3:  8,   68,   230,   387,    324,    108;
  4:  8,   84,   366,   847,   1098,    756,    216;
  5: 16,  192,   984,  2792,   4737,   4806,   2700,    648;
  6: 16,  224,  1368,  4760,  10321,  14280,  12312,   6048,   1296;
  7: 32,  496,  3408, 13624,  34922,  59523,  67464,  49032,  20736,   3888;
  8: 32,  560,  4400, 20440,  62170, 129367, 186510, 183960, 118800,  45360,   7776;
  9: 64, 1216, 10480, 54080, 185660, 445244, 761121, 927450, 789480, 447120, 151632, 23328;

The row polynomials satisfy P(n,x) = -(7*n+5+(19*n+16)*x+(12*n+12)*x^2)*P(n-1,x) + (4+18*x+26*x^2+12*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 7*x + 6*x^2.

P[0, x_] := 2 + 7*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[-(7*n + 5 + (19*n + 16)*x + (12*n + 12)*x^2)*P[n - 1, x] + (4 + 18*x + 26*x^2 + 12*x^3)*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] = 2 + 7*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = -(7*n + 5 + (19*n + 16)*x + (12*n + 12)*x^2)*v[i - 1] + (4 + 18*x + 26*x^2 + 12*x^3)*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 (−∞, 0]
  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 basis_map_A007318 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…?
    • Geometric basis map (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…?
            • 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)
    • 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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Candidate equalities ?

• A064861 → reverse → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A103633 → basis A013609 → x ↦ x+1
• A064861 → 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.