basis-map coefficients of A112227 → basis A013609 → (θ+1)²

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Submission text ?

Triangle begins:
   1: 1, 16,   36;
   2: 1, 24,  108,   128;
   3: 1, 32,  216,   512,    400;
   4: 1, 40,  360,  1280,   2000,   1152;
   5: 1, 48,  540,  2560,   6000,   6912,    3136;
   6: 1, 56,  756,  4480,  14000,  24192,   21952,    8192;
   7: 1, 64, 1008,  7168,  28000,  64512,   87808,   65536,   20736;
   8: 1, 72, 1296, 10752,  50400, 145152,  263424,  294912,  186624,   51200;
   9: 1, 80, 1620, 15360,  84000, 290304,  658560,  983040,  933120,  512000,  123904;
  10: 1, 88, 1980, 21120, 132000, 532224, 1448832, 2703360, 3421440, 2816000, 1362944, 294912;

The row polynomials satisfy (1+(6*n+4)*x+(4*n^2+8*n+4)*x^2)*P(n,x) = (1+(6*n+12)*x+(4*n^2+28*n+36)*x^2+(8*n^2+32*n+32)*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 16*x + 36*x^2.

P[1, x_] := 1 + 16*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (6*n + 12)*x + (4*n^2 + 28*n + 36)*x^2 + (8*n^2 + 32*n + 32)*x^3)*P[n - 1, x])/(1 + (6*n + 4)*x + (4*n^2 + 8*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 16*x + 36*x^2; for(n = 2, N, my(i = n); v[i] = ((1 + (6*n + 12)*x + (4*n^2 + 28*n + 36)*x^2 + (8*n^2 + 32*n + 32)*x^3)*v[i - 1])/(1 + (6*n + 4)*x + (4*n^2 + 8*n + 4)*x^2)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.?
    • Geometric basis map (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) 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)
        • 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 ?

• A130123 → basis A013609 → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A013609 → (θ+1)²
• A156992 → basis A013611 → (θ+1)²
• A178820 → basis A013611 → (θ+1)²
• A120909 → basis A013610 → (θ+1)²
• A178821 → basis A013611 → (θ+1)²
• A196347 → basis A013611 → (θ+1)²
• A120909 → (θ+1)²
• basis-map coefficients of A030528, shifted → basis A013609 → (θ+1)²
• A013609 → D → (θ+1)²
• A085880 → basis A013611 → (θ+1)²
• A028326 → basis A013611 → (θ+1)²
• A013609 → basis A013610 → (θ+1)²
• A013609 → (θ+1)²
• A038221 → basis A013611 → (θ+1)²
• A085881 → basis A013611 → (θ+1)²
• basis-map coefficients of A026729 → basis A013609 → (θ+1)²
• A136572 → basis A013609 → (θ+1)²
• A178822 → basis A013611 → (θ+1)²
• A356546 → basis A013611 → (θ+1)²
• A038208 → basis A013611 → (θ+1)²

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.