A008292 → γ (scale 1/4) (shift +1) → D

OEIS matches ?

• A090672: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:      0;
  1:      0;
  2:      2;
  3:     16;
  4:     88,      32;
  5:    416,     544;
  6:   1824,    5760,     816;
  7:   7680,   49152,   23808;
  8:  31616,  371712,  411648,   31744;
  9: 128512, 2609664, 5523456, 1415168;

The row polynomials satisfy (n-2)*P(n,x) = -(8*n-8+(4*n^2-20*n+16)*x+(n^3-7*n^2+14*n-8)*x^2)*P(n-2,x) - ((4*n-4)*x+(2*n^2-10*n+8)*x^2-(2*n^2-6*n+4)*x^3)*d/dx P(n-2,x) + (6*n-10+(2*n^2-10*n+12)*x)*P(n-1,x) + ((2*n-4)*x-(2*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 8 + (4*n^2 - 20*n + 16)*x + (n^3 - 7*n^2 + 14*n - 8)*x^2)*P[n - 2, x] - ((4*n - 4)*x + (2*n^2 - 10*n + 8)*x^2 - (2*n^2 - 6*n + 4)*x^3)*D[P[n - 2, x], x] + (6*n - 10 + (2*n^2 - 10*n + 12)*x)*P[n - 1, x] + ((2*n - 4)*x - (2*n - 4)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(8*n - 8 + (4*n^2 - 20*n + 16)*x + (n^3 - 7*n^2 + 14*n - 8)*x^2)*v[i - 2] - ((4*n - 4)*x + (2*n^2 - 10*n + 8)*x^2 - (2*n^2 - 6*n + 4)*x^3)*deriv(v[i - 2], x) + (6*n - 10 + (2*n^2 - 10*n + 12)*x)*v[i - 1] + ((2*n - 4)*x - (2*n - 4)*x^2)*deriv(v[i - 1], x))/(n - 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-4f4a07a2cead1a74
• 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-4f4a07a2cead1a74
• 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-4f4a07a2cead1a74

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-4f4a07a2cead1a74

  Laguerre’s theorem on differential operators [SP14] 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) → D

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.