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

OEIS matches ?

No OEIS matches.

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

The row polynomials satisfy (n+x)*P(n,x) = -(n+1+(n+2)*x+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[(-(n + 1 + (n + 2)*x + x^2)*P[n - 1, x])/(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] = (-(n + 1 + (n + 2)*x + x^2)*v[i - 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-3b50ee0d45bb8d84
• 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-3b50ee0d45bb8d84
• 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-3b50ee0d45bb8d84

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 derivative_polynomial, shift 0 e-3b50ee0d45bb8d84

  Laguerre’s theorem on differential operators [SP14] 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.

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

References

1. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.