A227450 → 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:    2;
  1:    6,      4;
  2:   14,     28,       8;
  3:   30,    124,      96,      16;
  4:   62,    460,     680,     288,      32;
  5:  126,   1564,    3840,    2960,     800,      64;
  6:  254,   5068,   19208,   23520,   11200,    2112,     128;
  7:  510,  15964,   89376,  161616,  118720,   38528,    5376,    256;
  8: 1022,  49420,  397160, 1014048, 1069152,  524160,  123648,  13312,   512;
  9: 2046, 151324, 1711680, 5988880, 8665440, 5976768, 2096640, 376320, 32256, 1024;

The row polynomials satisfy P(n,x) = -(2+3*x+2*x^2)*P(n-2,x) - (x+x^2)*d/dx P(n-2,x) + (3+3*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 6 + 4*x.

P[0, x_] := 2;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 3*x + 2*x^2)*P[n - 2, x] - (x + x^2)*D[P[n - 2, x], x] + (3 + 3*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 3*x + 2*x^2)*v[i - 2] - (x + x^2)*deriv(v[i - 2], x) + (3 + 3*x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-b0ca16d4a4893249
• 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-b0ca16d4a4893249
• 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-b0ca16d4a4893249
• 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-b0ca16d4a4893249

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-b0ca16d4a4893249

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

  • premise (sequence): Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-74361f30d2bc482d root: theorem check

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

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.