basis-map coefficients of A026729 → basis A007318 → (θ+1)²

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Triangle begins:
   1: 1;
   2: 1,  4;
   3: 1,  8,   9;
   4: 1, 12,  27,   16;
   5: 1, 16,  54,   64,   25;
   6: 1, 20,  90,  160,  125,   36;
   7: 1, 24, 135,  320,  375,  216,   49;
   8: 1, 28, 189,  560,  875,  756,  343,   64;
   9: 1, 32, 252,  896, 1750, 2016, 1372,  512,  81;
  10: 1, 36, 324, 1344, 3150, 4536, 4116, 2304, 729, 100;

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

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

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + 3*n*x + (n^2 + 3*n - 1)*x^2 + n^2*x^3)*v[i - 1])/(1 + (3*n - 4)*x + (n^2 - 2*n + 1)*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 shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
        • 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 basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
            • 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
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
            • 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
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A038325 → basis A013609 → (θ+1)²
• A178820 → (θ+1)²
• A156992 → basis A013609 → (θ+1)²
• A085881 → (θ+1)²
• A003506 → D → (θ+1)²
• A038338 → (θ+1)²
• A178822 → basis A013609 → (θ+1)²
• A003506 → (θ+1)²
• A028326 → (θ+1)²
• basis-map coefficients of A127647 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• A125092 → reverse
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A344565 → (θ+1)²
• A135065 → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• A134346 → (θ+1)²
• A356546 → basis A013609 → (θ+1)²
• A085880 → basis A013609 → (θ+1)²
• A085880 → D → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A085881 → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A178822 → (θ+1)²
• A038221 → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
• A196347 → basis A013609 → (θ+1)²
• A028326 → D → (θ+1)²
• A196347 → (θ+1)²
• A156992 → (θ+1)²
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
• A178821 → (θ+1)²
• A038338 → basis A013609 → (θ+1)²
• A085880 → (θ+1)²
• A376827 → (θ+1)²
• A124860 → (θ+1)²
• A038208 → D → (θ+1)²
• A125092 → reverse → D
• A038330 → basis A013611 → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• basis-map coefficients of A113214, shifted → D → (θ+1)²
• A135838 → (θ+1)²
• A356546 → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• A038325 → (θ+1)²
• A120910 → basis A013610 → (θ+1)²
• A038208 → (θ+1)²
• A178820 → basis A013609 → (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.