A026374 → D+1 → x ↦ x+1

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Triangle begins:
  0:    1;
  1:    3,     1;
  2:   10,     7,     1;
  3:   25,    29,    10,     1;
  4:   75,   120,    65,    14,     1;
  5:  175,   365,   285,   103,    17,     1;
  6:  500,  1275,  1275,   635,   165,    21,    1;
  7: 1125,  3425,  4275,  2820,  1055,   222,   24,   1;
  8: 3125, 11000, 16250, 13100,  6275,  1820,  310,  28,  1;
  9: 6875, 27625, 47750, 46450, 27925, 10715, 2610, 386, 31, 1;

The row polynomials satisfy P(n,x) = -(25+50*x+35*x^2+10*x^3+x^4)*P(n-4,x) + (10+10*x+2*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 10 + 7*x + x^2, P(3,x) = 25 + 29*x + 10*x^2 + x^3.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 10 + 7*x + x^2;
P[3, x_] := 25 + 29*x + 10*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(25 + 50*x + 35*x^2 + 10*x^3 + x^4)*P[n - 4, x] + (10 + 10*x + 2*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 10 + 7*x + x^2; v[4] = 25 + 29*x + 10*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = -(25 + 50*x + 35*x^2 + 10*x^3 + x^4)*v[i - 4] + (10 + 10*x + 2*x^2)*v[i - 2]); 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 one_plus_derivative of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1…?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)

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.