A062190 → D

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:   0;
  1:   6;
  2:  14,   42;
  3:  24,  168,   168;
  4:  36,  432,  1008,    504;
  5:  50,  900,  3600,   4200,    1260;
  6:  66, 1650,  9900,  19800,   13860,    2772;
  7:  84, 2772, 23100,  69300,   83160,   38808,   5544;
  8: 104, 4368, 48048, 200200,  360360,  288288,  96096,  10296;
  9: 126, 6552, 91728, 504504, 1261260, 1513512, 864864, 216216, 18018;

The row polynomials satisfy (n^2+3*n-4)*P(n,x) = (n^2+5*n+(3*n^2+9*n+12)*x)*P(n-1,x) + ((2*n+4)*x-(2*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.

P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + (3*n^2 + 9*n + 12)*x)*P[n - 1, x] + ((2*n + 4)*x - (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 + 3*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 5*n + (3*n^2 + 9*n + 12)*x)*v[i - 1] + ((2*n + 4)*x - (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 + 3*n - 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-581332e7358bcae7
• 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-581332e7358bcae7
• 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-581332e7358bcae7
• 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-581332e7358bcae7

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-581332e7358bcae7

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

  • premise (sequence): Coefficient triangle of certain polynomials N(5; m,x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-e968c429d4b85e0b 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.