A304236 → (θ+1)²

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: 1, 12;
  3: 1, 24;
  4: 1, 36,   81;
  5: 1, 48,  243;
  6: 1, 60,  486,  432;
  7: 1, 72,  810, 1728;
  8: 1, 84, 1215, 4320,  2025;
  9: 1, 96, 1701, 8640, 10125;

The row polynomials satisfy (n^2+n+(9*n^3-6*n)*x+(9*n^4+9*n^3)*x^2)*P(n,x) = (n^2+n+(9*n^3+24*n^2-6*n-24)*x+(9*n^4+99*n^3+72*n^2-126*n-36)*x^2+(54*n^4+216*n^3+162*n^2-216*n-216)*x^3)*P(n-1,x) - ((n+1)*x+(9*n^2+30*n+18)*x^2+(9*n^3+153*n^2+288*n+108)*x^3+(108*n^3+540*n^2+864*n+432)*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[((n^2 + n + (9*n^3 + 24*n^2 - 6*n - 24)*x + (9*n^4 + 99*n^3 + 72*n^2 - 126*n - 36)*x^2 + (54*n^4 + 216*n^3 + 162*n^2 - 216*n - 216)*x^3)*P[n - 1, x] - ((n + 1)*x + (9*n^2 + 30*n + 18)*x^2 + (9*n^3 + 153*n^2 + 288*n + 108)*x^3 + (108*n^3 + 540*n^2 + 864*n + 432)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (9*n^3 - 6*n)*x + (9*n^4 + 9*n^3)*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] = ((n^2 + n + (9*n^3 + 24*n^2 - 6*n - 24)*x + (9*n^4 + 99*n^3 + 72*n^2 - 126*n - 36)*x^2 + (54*n^4 + 216*n^3 + 162*n^2 - 216*n - 216)*x^3)*v[i - 1] - ((n + 1)*x + (9*n^2 + 30*n + 18)*x^2 + (9*n^3 + 153*n^2 + 288*n + 108)*x^3 + (108*n^3 + 540*n^2 + 864*n + 432)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (9*n^3 - 6*n)*x + (9*n^4 + 9*n^3)*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-4e4cfdea6d7e0f6f
• 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-4e4cfdea6d7e0f6f
• 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-4e4cfdea6d7e0f6f

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-4e4cfdea6d7e0f6f

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.