A008292 → (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,    4;
   3: 1,   16,      9;
   4: 1,   44,     99,      16;
   5: 1,  104,    594,     416,       25;
   6: 1,  228,   2718,    4832,     1425,       36;
   7: 1,  480,  10719,   38656,    29775,     4320,       49;
   8: 1,  988,  38637,  249904,   390475,   154548,    12103,      64;
   9: 1, 2008, 131472, 1411744,  3904750,  3176424,   715792,   32128,    81;
  10: 1, 4052, 430560, 7283072, 32758850, 47172744, 22304408, 3061760, 82053, 100;

The row polynomials satisfy (n-1)*P(n,x) = -((4*n^2-8*n)*x-(9*n^3-29*n^2+24*n-4)*x^2)*P(n-2,x) - (2*n*x-(4*n^2+1)*x^2-(4*n^3-26*n^2+27*n-5)*x^3)*d/dx P(n-2,x) - (n*x^2-(4*n^2-4*n+1)*x^3+(4*n^2-5*n+1)*x^4)*d^2/dx^2 P(n-2,x) + (n-1+(4*n^2-8*n+4)*x)*P(n-1,x) + ((2*n-1)*x-(4*n^2-4*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 - 8*n)*x - (9*n^3 - 29*n^2 + 24*n - 4)*x^2)*P[n - 2, x] - (2*n*x - (4*n^2 + 1)*x^2 - (4*n^3 - 26*n^2 + 27*n - 5)*x^3)*D[P[n - 2, x], x] - (n*x^2 - (4*n^2 - 4*n + 1)*x^3 + (4*n^2 - 5*n + 1)*x^4)*D[P[n - 2, x], {x, 2}] + (n - 1 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] + ((2*n - 1)*x - (4*n^2 - 4*n)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = (-((4*n^2 - 8*n)*x - (9*n^3 - 29*n^2 + 24*n - 4)*x^2)*v[i - 2] - (2*n*x - (4*n^2 + 1)*x^2 - (4*n^3 - 26*n^2 + 27*n - 5)*x^3)*deriv(v[i - 2], x) - (n*x^2 - (4*n^2 - 4*n + 1)*x^3 + (4*n^2 - 5*n + 1)*x^4)*deriv(deriv(v[i - 2], x), x) + (n - 1 + (4*n^2 - 8*n + 4)*x)*v[i - 1] + ((2*n - 1)*x - (4*n^2 - 4*n)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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-3a7fca57ac79b222
• 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-3a7fca57ac79b222
• 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-3a7fca57ac79b222
• 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-3a7fca57ac79b222

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-3a7fca57ac79b222

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

• A173018 → (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.