A008292 → γ (shift +1) → (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,    8;
  3: 1,   32;
  4: 1,   88,    144;
  5: 1,  208,   1224;
  6: 1,  456,   6480,    4352;
  7: 1,  960,  27648,   63488;
  8: 1, 1976, 104544,  548864,  198400;
  9: 1, 4016, 366984, 3682304, 4422400;

The row polynomials satisfy n*P(n,x) = -((8*n^2-8*n-16)*x-(36*n^3-52*n^2-40*n)*x^2)*P(n-2,x) - ((2*n+2)*x-(8*n^2+24*n+20)*x^2-(16*n^3-128*n^2-72*n)*x^3)*d/dx P(n-2,x) - ((n+1)*x^2-(8*n^2+8*n+4)*x^3+(32*n^2+16*n)*x^4)*d^2/dx^2 P(n-2,x) + (n+(8*n^2-8*n)*x)*P(n-1,x) + ((2*n+1)*x-(8*n^2+8*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^2 - 8*n - 16)*x - (36*n^3 - 52*n^2 - 40*n)*x^2)*P[n - 2, x] - ((2*n + 2)*x - (8*n^2 + 24*n + 20)*x^2 - (16*n^3 - 128*n^2 - 72*n)*x^3)*D[P[n - 2, x], x] - ((n + 1)*x^2 - (8*n^2 + 8*n + 4)*x^3 + (32*n^2 + 16*n)*x^4)*D[P[n - 2, x], {x, 2}] + (n + (8*n^2 - 8*n)*x)*P[n - 1, x] + ((2*n + 1)*x - (8*n^2 + 8*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^2 - 8*n - 16)*x - (36*n^3 - 52*n^2 - 40*n)*x^2)*v[i - 2] - ((2*n + 2)*x - (8*n^2 + 24*n + 20)*x^2 - (16*n^3 - 128*n^2 - 72*n)*x^3)*deriv(v[i - 2], x) - ((n + 1)*x^2 - (8*n^2 + 8*n + 4)*x^3 + (32*n^2 + 16*n)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + (8*n^2 - 8*n)*x)*v[i - 1] + ((2*n + 1)*x - (8*n^2 + 8*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-019ad329539bbbb3
• 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-019ad329539bbbb3
• 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-019ad329539bbbb3

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-019ad329539bbbb3

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

  • premise (sequence): Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1, shift 1 e-e49ea542bfed6bbf

      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 ?

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.