A008292 → γ (scale 1/4) (shift +1) → (θ+1)²

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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:   2;
  2:   4,      8;
  3:   8,     64;
  4:  16,    352,      144;
  5:  32,   1664,     2448;
  6:  64,   7296,    25920,     4352;
  7: 128,  30720,   221184,   126976;
  8: 256, 126464,  1672704,  2195456,  198400;
  9: 512, 514048, 11743488, 29458432, 8844800;

The row polynomials satisfy n*P(n,x) = -((8*n^2-8*n-16)*x-(9*n^3-13*n^2-10*n)*x^2)*P(n-2,x) - ((8*n+8)*x-(8*n^2+24*n+20)*x^2-(4*n^3-32*n^2-18*n)*x^3)*d/dx P(n-2,x) - ((4*n+4)*x^2-(8*n^2+8*n+4)*x^3+(8*n^2+4*n)*x^4)*d^2/dx^2 P(n-2,x) + (2*n+(4*n^2-4*n)*x)*P(n-1,x) + ((4*n+2)*x-(4*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^2 - 8*n - 16)*x - (9*n^3 - 13*n^2 - 10*n)*x^2)*P[n - 2, x] - ((8*n + 8)*x - (8*n^2 + 24*n + 20)*x^2 - (4*n^3 - 32*n^2 - 18*n)*x^3)*D[P[n - 2, x], x] - ((4*n + 4)*x^2 - (8*n^2 + 8*n + 4)*x^3 + (8*n^2 + 4*n)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n + (4*n^2 - 4*n)*x)*P[n - 1, x] + ((4*n + 2)*x - (4*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^2 - 8*n - 16)*x - (9*n^3 - 13*n^2 - 10*n)*x^2)*v[i - 2] - ((8*n + 8)*x - (8*n^2 + 24*n + 20)*x^2 - (4*n^3 - 32*n^2 - 18*n)*x^3)*deriv(v[i - 2], x) - ((4*n + 4)*x^2 - (8*n^2 + 8*n + 4)*x^3 + (8*n^2 + 4*n)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n + (4*n^2 - 4*n)*x)*v[i - 1] + ((4*n + 2)*x - (4*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n)); 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-e126124009b7b91d
• 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-e126124009b7b91d
• 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-e126124009b7b91d

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-e126124009b7b91d

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

  • premise (sequence): Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1/4, shift 1 e-00ceaf3777abe095

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): eulerian
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-ab3cdcfae289b48d root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • parent_provers root_domain e-6c46c948823f0a8c root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-c61ecf4ef1c45372

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): eulerian
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-ab3cdcfae289b48d (expanded above)
            • ↵ parent_provers root_domain e-6c46c948823f0a8c (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A008292 → γ (scale 1/4) → (θ+1)²

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.