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

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OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:  1;
  1: -1,  -2;
  2:  1,   4,    3;
  3: -1,  -6,   -9,   -4;
  4:  1,   8,   18,   16,    5;
  5: -1, -10,  -30,  -40,  -25,   -6;
  6:  1,  12,   45,   80,   75,   36,    7;
  7: -1, -14,  -63, -140, -175, -126,  -49,   -8;
  8:  1,  16,   84,  224,  350,  336,  196,   64,   9;
  9: -1, -18, -108, -336, -630, -756, -588, -288, -81, -10;

The row polynomials satisfy (1+n*x)*P(n,x) = -(1+(n+2)*x+(n+1)*x^2)*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 + (n + 2)*x + (n + 1)*x^2)*P[n - 1, x])/(1 + n*x)]];
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 + (n + 2)*x + (n + 1)*x^2)*v[i - 1])/(1 + n*x)); 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-dfc204a5cd0fb4b5
• 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-dfc204a5cd0fb4b5
• 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-dfc204a5cd0fb4b5

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-dfc204a5cd0fb4b5

  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 (27)

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

• A038208 → θ+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A007318 → ×(1+2n+n²) → D+1
• A178821 → D+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A007318 → ×(1+(3/2)n+(1/2)n²) → D+1
• A085880 → θ+1
• A007318 → D → D+1
• A007318 → D → θ+1
• A135838 → D+1
• A178821 → θ+1
• A344565 → D+1
• A344565 → θ+1
• A135838 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• basis-map coefficients of A113214, shifted → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A178822 → D+1
• A178820 → θ+1
• A178820 → D+1
• A085880 → D+1
• A028326 → θ+1
• A178822 → θ+1
• A028326 → D+1
• A007318 → ×(1+2n+n²) → θ+1
• A038208 → D+1

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.