A173018 → reverse (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: 0, 4;
  2: 0, 4,    9;
  3: 0, 4,   36,     16;
  4: 0, 4,   99,    176,      25;
  5: 0, 4,  234,   1056,     650,      36;
  6: 0, 4,  513,   4832,    7550,    2052,      49;
  7: 0, 4, 1080,  19056,   60400,   42876,    5880,     64;
  8: 0, 4, 2223,  68688,  390475,  562284,  210357,  15808,    81;
  9: 0, 4, 4518, 233728, 2205850, 5622840, 4323466, 934912, 40662, 100;

The row polynomials satisfy n^2*P(n,x) = ((4*n-4)*x+(9*n^4-2*n^3-7*n^2)*x^2)*P(n-2,x) + (x+(4*n^2+5*n-5)*x^2+(4*n^4-14*n^3-13*n^2)*x^3)*d/dx P(n-2,x) - ((n^2+n-1)*x^2-(4*n^3+4*n^2+n-1)*x^3+(4*n^3+3*n^2)*x^4)*d^2/dx^2 P(n-2,x) - (n^2+n-4*n^3*x)*P(n-1,x) + ((2*n^2+n-1)*x-(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.

P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(((4*n - 4)*x + (9*n^4 - 2*n^3 - 7*n^2)*x^2)*P[n - 2, x] + (x + (4*n^2 + 5*n - 5)*x^2 + (4*n^4 - 14*n^3 - 13*n^2)*x^3)*D[P[n - 2, x], x] - ((n^2 + n - 1)*x^2 - (4*n^3 + 4*n^2 + n - 1)*x^3 + (4*n^3 + 3*n^2)*x^4)*D[P[n - 2, x], {x, 2}] - (n^2 + n - 4*n^3*x)*P[n - 1, x] + ((2*n^2 + n - 1)*x - (4*n^3 + 4*n^2)*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] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((4*n - 4)*x + (9*n^4 - 2*n^3 - 7*n^2)*x^2)*v[i - 2] + (x + (4*n^2 + 5*n - 5)*x^2 + (4*n^4 - 14*n^3 - 13*n^2)*x^3)*deriv(v[i - 2], x) - ((n^2 + n - 1)*x^2 - (4*n^3 + 4*n^2 + n - 1)*x^3 + (4*n^3 + 3*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) - (n^2 + n - 4*n^3*x)*v[i - 1] + ((2*n^2 + n - 1)*x - (4*n^3 + 4*n^2)*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-6aabb2282cf76ea5
• 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-6aabb2282cf76ea5
• 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-6aabb2282cf76ea5

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-6aabb2282cf76ea5

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

  • premise (sequence): Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 1 e-61bc78adfe1eb25a

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-083b494dcfd2f637 root: theorem check

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

        • parent_provers root_domain e-ffc8a9fffd6e96cb root: theorem check

          Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A334218 → (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.