A173018 → (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:
   1: 1;
   2: 1;
   3: 1,    4;
   4: 1,   16,      9;
   5: 1,   44,     99,      16;
   6: 1,  104,    594,     416,      25;
   7: 1,  228,   2718,    4832,    1425,      36;
   8: 1,  480,  10719,   38656,   29775,    4320,     49;
   9: 1,  988,  38637,  249904,  390475,  154548,  12103,    64;
  10: 1, 2008, 131472, 1411744, 3904750, 3176424, 715792, 32128, 81;

The row polynomials satisfy (n-2)*P(n,x) = -((4*n^2-16*n+12)*x-(9*n^3-56*n^2+109*n-66)*x^2)*P(n-2,x) - ((2*n-2)*x-(4*n^2-8*n+5)*x^2-(4*n^3-38*n^2+91*n-62)*x^3)*d/dx P(n-2,x) - ((n-1)*x^2-(4*n^2-12*n+9)*x^3+(4*n^2-13*n+10)*x^4)*d^2/dx^2 P(n-2,x) + (n-2+(4*n^2-16*n+16)*x)*P(n-1,x) + ((2*n-3)*x-(4*n^2-12*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 - 16*n + 12)*x - (9*n^3 - 56*n^2 + 109*n - 66)*x^2)*P[n - 2, x] - ((2*n - 2)*x - (4*n^2 - 8*n + 5)*x^2 - (4*n^3 - 38*n^2 + 91*n - 62)*x^3)*D[P[n - 2, x], x] - ((n - 1)*x^2 - (4*n^2 - 12*n + 9)*x^3 + (4*n^2 - 13*n + 10)*x^4)*D[P[n - 2, x], {x, 2}] + (n - 2 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] + ((2*n - 3)*x - (4*n^2 - 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-((4*n^2 - 16*n + 12)*x - (9*n^3 - 56*n^2 + 109*n - 66)*x^2)*v[i - 2] - ((2*n - 2)*x - (4*n^2 - 8*n + 5)*x^2 - (4*n^3 - 38*n^2 + 91*n - 62)*x^3)*deriv(v[i - 2], x) - ((n - 1)*x^2 - (4*n^2 - 12*n + 9)*x^3 + (4*n^2 - 13*n + 10)*x^4)*deriv(deriv(v[i - 2], x), x) + (n - 2 + (4*n^2 - 16*n + 16)*x)*v[i - 1] + ((2*n - 3)*x - (4*n^2 - 12*n + 8)*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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-55c66a5e0fb07f9a
• Coefficients of one sign. Every nonzero row, n >= 1: 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-55c66a5e0fb07f9a
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-55c66a5e0fb07f9a
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-55c66a5e0fb07f9a

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-55c66a5e0fb07f9a

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

• A008292 → (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.