basis-map coefficients of A112227 → x ↦ x+1 → D+1

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Triangle begins:
   1:  3,   4,   1;
   2:  4,   9,   6,    1;
   3:  5,  16,  18,    8,    1;
   4:  6,  25,  40,   30,   10,    1;
   5:  7,  36,  75,   80,   45,   12,    1;
   6:  8,  49, 126,  175,  140,   63,   14,    1;
   7:  9,  64, 196,  336,  350,  224,   84,   16,   1;
   8: 10,  81, 288,  588,  756,  630,  336,  108,  18,   1;
   9: 11, 100, 405,  960, 1470, 1512, 1050,  480, 135,  20,  1;
  10: 12, 121, 550, 1485, 2640, 3234, 2772, 1650, 660, 165, 22, 1;

The row polynomials satisfy (n+1+x)*P(n,x) = (n+2+(n+3)*x+x^2)*P(n-1,x) for n >= 2, with P(1,x) = 3 + 4*x + x^2.

P[1, x_] := 3 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (n + 3)*x + x^2)*P[n - 1, x])/(n + 1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 3 + 4*x + x^2; for(n = 2, N, my(i = n); v[i] = ((n + 2 + (n + 3)*x + x^2)*v[i - 1])/(n + 1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • Parent-row provers of polytool polytool ? step › direct check
        • 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 basis-map coefficients of A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A027465 → basis A013611 → θ+1
• A085880 → D → D+1
• basis-map coefficients of A026729 → basis A007318 → D+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A038338 → basis A013609 → D+1
• A196347 → basis A013609 → D+1
• A038208 → D → D+1
• A120910 → basis A013610 → θ+1
• A038221 → basis A013609 → θ+1
• A344565 → D+1
• A085880 → D → θ+1
• A085880 → D+1
• A038221 → basis A013609 → D+1
• A356546 → θ+1
• A038221 → θ+1
• A094305 → θ+1
• A038325 → basis A013609 → D+1
• A344565 → θ+1
• A178820 → basis A013609 → D+1
• A094305 → D+1
• A124860 → D+1
• A085880 → θ+1
• A038208 → θ+1
• basis-map coefficients of A127647 → θ+1
• A093375 → D → D
• A178821 → basis A013609 → θ+1
• A085881 → basis A013609 → θ+1
• A027465 → basis A013611 → D+1
• A216973 → θ+1
• A028326 → D → θ+1
• A131431 → x ↦ x+1 → θ+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A038338 → basis A013609 → θ+1
• A028326 → θ+1
• A085881 → basis A013609 → D+1
• A178822 → θ+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• A376827 → θ+1
• A085880 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• A178821 → θ+1
• A135838 → D+1
• A028326 → basis A013609 → θ+1
• A178820 → basis A013609 → θ+1
• A178820 → θ+1
• A028326 → basis A013609 → D+1
• A038330 → basis A013611 → θ+1
• A085880 → basis A013609 → D+1
• A178822 → basis A013609 → θ+1
• A156992 → D+1
• A038325 → basis A013609 → θ+1
• A135065 → D+1
• A135065 → θ+1
• A156992 → basis A013609 → D+1
• A120910 → basis A013610 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A038325 → θ+1
• A028326 → D → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A156992 → basis A013609 → θ+1
• A135838 → θ+1
• A131431 → D+1 → basis A007318
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A196347 → D+1
• A134346 → D+1
• A178822 → basis A013609 → D+1
• A178821 → basis A013609 → D+1
• A038330 → basis A013611 → D+1
• A124860 → θ+1
• A131431 → x ↦ x+1 → D+1
• basis-map coefficients of A113214, shifted → D+1
• basis-map coefficients of A113214, shifted → θ+1
• basis-map coefficients of A113214, shifted → D → D+1
• basis-map coefficients of A127647 → D+1
• A038221 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A356546 → D+1
• A134346 → θ+1
• A085881 → θ+1
• A038338 → θ+1
• A156992 → θ+1
• A356546 → basis A013609 → θ+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A376827 → D+1
• A178822 → D+1
• A038325 → D+1
• A038208 → D → θ+1
• A196347 → θ+1
• A178821 → D+1
• A038208 → D+1
• A093375 → D
• A178820 → D+1
• A028326 → D+1
• A003506 → D → θ+1
• A196347 → basis A013609 → θ+1
• A038338 → D+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A356546 → basis A013609 → D+1
• A085881 → D+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.