A152198 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:  0;
  1:  0;
  2:  1;
  3:  1;
  4:  4,  2;
  5:  4,  2;
  6: 12, 12,  3;
  7: 12, 12,  3;
  8: 32, 48, 24, 4;
  9: 32, 48, 24, 4;

The row polynomials satisfy (n-2)*P(n,x) = (n^2-8+(n^2-3*n)*x)*P(n-1,x) - (4*n+8+(6*n+4)*x+2*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 8 + (n^2 - 3*n)*x)*P[n - 1, x] - (4*n + 8 + (6*n + 4)*x + 2*n*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 - 8 + (n^2 - 3*n)*x)*v[i - 1] - (4*n + 8 + (6*n + 4)*x + 2*n*x^2)*deriv(v[i - 1], x))/(n - 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 (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from shift_x_plus_one of Triangle read by rows, A007318 rows repeated.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, A007318 rows repeated.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
            • 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
        • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? 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)
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows, A007318 rows repeated.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Gamma criterion (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A274498 → D → D
• A274496 → basis A013611
• A274498: Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k…
• A274498 → D
• A152198 → basis A013611
• A188440: Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
• A152198 → basis A013611 → basis A013610
• A152198 → basis A013611 → D
• A274498 → basis A013610
• A188440 → basis A013610
• A274496 → basis A013611 → D
• A274496 → x ↦ x+1
• A274496 → basis A013611 → basis A013610
• A152198 → x ↦ x+1
• A152198 → basis A013609 → basis A007318
• A274496 → D → x ↦ x+1
• A188440 → D
• A152198 → basis A013609 → basis A013611
• A103633 → 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.