basis-map coefficients of A113214, shifted → (θ+1)²

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Triangle begins:
  0:  1;
  1: -1,  -4;
  2:  1,   8,    9;
  3: -1, -12,  -27,   -16;
  4:  1,  16,   54,    64,    25;
  5: -1, -20,  -90,  -160,  -125,   -36;
  6:  1,  24,  135,   320,   375,   216,    49;
  7: -1, -28, -189,  -560,  -875,  -756,  -343,   -64;
  8:  1,  32,  252,   896,  1750,  2016,  1372,   512,   81;
  9: -1, -36, -324, -1344, -3150, -4536, -4116, -2304, -729, -100;

The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = -(1+(3*n+3)*x+(n^2+5*n+3)*x^2+(n^2+2*n+1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + (3*n + 3)*x + (n^2 + 5*n + 3)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(1 + (3*n + 3)*x + (n^2 + 5*n + 3)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*x^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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-0045543824a8afca
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-0045543824a8afca
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-0045543824a8afca

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image euler_polynomial, shift 0 e-0045543824a8afca

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-60aa79917842b999

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • basis_map A007318 e-c62e3171974f4f9c

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers periodic_product e-5a4f04ff49b735c1 root: theorem check

              Parent-row provers of polytool polytool ?

            • preserver_image mobius, shift 0 e-4adca88d3fcf673e

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ parent_provers periodic_product e-5a4f04ff49b735c1 (expanded above)
        • parent_provers periodic_product e-c76ca1e6cd597d3b root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image affine, shift 0 e-c58069640929a62b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-5a4f04ff49b735c1 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-4adca88d3fcf673e (expanded above)
    • ↵ basis_map A007318 e-c62e3171974f4f9c (expanded above)
    • ↵ parent_provers periodic_product e-c76ca1e6cd597d3b (expanded above)
    • ↵ preserver_image affine, shift 0 e-c58069640929a62b (expanded above)

Related sequences ?

Same fingerprint (12)

Equal up to content, sign, reversal and a small index shift.

• A085880 → (θ+1)²
• A135838 → (θ+1)²
• A178821 → (θ+1)²
• A178822 → (θ+1)²
• A178820 → (θ+1)²
• A038208 → (θ+1)²
• A344565 → (θ+1)²
• A007318 → ×(1+2n+n²) → (θ+1)²
• A007318 → (θ+1)² → reverse
• A007318 → D → (θ+1)²
• A028326 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.