A304236 → 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:  1;
  2:  4,   3;
  3:  7,   6;
  4: 10,  27,    9;
  5: 13,  66,   27;
  6: 16, 123,  135,   27;
  7: 19, 198,  414,  108;
  8: 22, 291,  945,  594,  81;
  9: 25, 402, 1809, 2160, 405;

The row polynomials satisfy (3*n^3-8*n^2+5*n+(12*n^2-29*n)*x+12*n*x^2)*P(n,x) = (3*n^3-5*n^2+2*n+(18*n^3-54*n^2+19*n)*x+(72*n^2-162*n+102)*x^2+(72*n-72)*x^3)*P(n-1,x) - ((3*n^2-2*n)*x+(36*n^2-12*n-5)*x^2+(144*n-48)*x^3+144*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 5*n^2 + 2*n + (18*n^3 - 54*n^2 + 19*n)*x + (72*n^2 - 162*n + 102)*x^2 + (72*n - 72)*x^3)*P[n - 1, x] - ((3*n^2 - 2*n)*x + (36*n^2 - 12*n - 5)*x^2 + (144*n - 48)*x^3 + 144*x^4)*D[P[n - 1, x], x])/(3*n^3 - 8*n^2 + 5*n + (12*n^2 - 29*n)*x + 12*n*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] = ((3*n^3 - 5*n^2 + 2*n + (18*n^3 - 54*n^2 + 19*n)*x + (72*n^2 - 162*n + 102)*x^2 + (72*n - 72)*x^3)*v[i - 1] - ((3*n^2 - 2*n)*x + (36*n^2 - 12*n - 5)*x^2 + (144*n - 48)*x^3 + 144*x^4)*deriv(v[i - 1], x))/(3*n^3 - 8*n^2 + 5*n + (12*n^2 - 29*n)*x + 12*n*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-8635ad6e77f468ea
• 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-8635ad6e77f468ea
• 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-8635ad6e77f468ea

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-8635ad6e77f468ea

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by ro…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers paper_theorem e-0050d566ad712b7d root: theorem check

      Parent-row provers of polytool polytool ?

    • gamma_polynomial scale 1, shift 0 e-64b74724257dbeaa

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Pascal-(1,3,1) array.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang_derivative_lag e-fba968684828f914 root: theorem check

          Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

        • preserver_image reversal, shift 0 e-86685874cd5f1179

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Pascal-(1,3,1) array.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang_derivative_lag e-fba968684828f914 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.