A249252 → D+1

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:  5,    4;
  2:  9,   24,     8;
  3: 13,   76,    80,    16;
  4: 17,  160,   360,   224,     32;
  5: 21,  276,  1040,  1328,    576,     64;
  6: 25,  424,  2312,  4928,   4256,   1408,    128;
  7: 29,  604,  4368, 13648,  19392,  12480,   3328,   256;
  8: 33,  816,  7400, 31136,  64800,  67584,  34432,  7680,   512;
  9: 37, 1060, 11600, 62064, 174912, 265536, 216320, 90880, 17408, 1024;

The row polynomials satisfy -(4*n^3+n^2-3*n-(8*n^3-10*n^2-3*n+15)*x-(20*n^2-38*n+10)*x^2-(8*n^2-4*n-12)*x^3-(8*n-8)*x^4)*P(n,x) = -(4*n^3+5*n^2+n+(24*n^3-6*n^2-45*n-15)*x+(44*n^2-34*n-10)*x^2+(24*n^2+36*n+12)*x^3+(24*n+8)*x^4)*P(n-1,x) + ((4*n^2+5*n+1)*x+(48*n^2+72*n+19)*x^2+(16*n^2+168*n+100)*x^3+(96*n+176)*x^4+(16*n+96)*x^5+16*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 + 5*n^2 + n + (24*n^3 - 6*n^2 - 45*n - 15)*x + (44*n^2 - 34*n - 10)*x^2 + (24*n^2 + 36*n + 12)*x^3 + (24*n + 8)*x^4)*P[n - 1, x] + ((4*n^2 + 5*n + 1)*x + (48*n^2 + 72*n + 19)*x^2 + (16*n^2 + 168*n + 100)*x^3 + (96*n + 176)*x^4 + (16*n + 96)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(-4*n^3 - n^2 + 3*n + (8*n^3 - 10*n^2 - 3*n + 15)*x + (20*n^2 - 38*n + 10)*x^2 + (8*n^2 - 4*n - 12)*x^3 + (8*n - 8)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n^3 + 5*n^2 + n + (24*n^3 - 6*n^2 - 45*n - 15)*x + (44*n^2 - 34*n - 10)*x^2 + (24*n^2 + 36*n + 12)*x^3 + (24*n + 8)*x^4)*v[i - 1] + ((4*n^2 + 5*n + 1)*x + (48*n^2 + 72*n + 19)*x^2 + (16*n^2 + 168*n + 100)*x^3 + (96*n + 176)*x^4 + (16*n + 96)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(-4*n^3 - n^2 + 3*n + (8*n^3 - 10*n^2 - 3*n + 15)*x + (20*n^2 - 38*n + 10)*x^2 + (8*n^2 - 4*n - 12)*x^3 + (8*n - 8)*x^4)); 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-65d44126837c3e43
• 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-65d44126837c3e43
• 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-65d44126837c3e43
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-65d44126837c3e43

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-65d44126837c3e43

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-fb471901eb718e65 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

No candidate equalities or related sequences.

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.