A121574 → 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:    1;
  2:    5,     2;
  3:   16,    16,     3;
  4:   44,    74,    33,     4;
  5:  112,   268,   201,    56,     5;
  6:  272,   848,   915,   424,    85,    6;
  7:  640,  2464,  3504,  2336,   770,  120,    7;
  8: 1472,  6752, 11976, 10564,  4990, 1266,  161,   8;
  9: 3328, 17728, 37776, 41768, 26105, 9444, 1939, 208, 9;

The row polynomials satisfy -(6*n^2-14*n+8-(3*n^2+n-4)*x+(n-1)*x^2)*P(n,x) = -(12*n^2-16*n+4+(12*n^2-50*n+10)*x-(13*n-11)*x^2+2*x^3)*P(n-1,x) + ((12*n-4)*x+(24*n-12)*x^2+(3*n-9)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n^2 - 16*n + 4 + (12*n^2 - 50*n + 10)*x - (13*n - 11)*x^2 + 2*x^3)*P[n - 1, x] + ((12*n - 4)*x + (24*n - 12)*x^2 + (3*n - 9)*x^3 - x^4)*D[P[n - 1, x], x])/(-6*n^2 + 14*n - 8 + (3*n^2 + n - 4)*x - (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^2 - 16*n + 4 + (12*n^2 - 50*n + 10)*x - (13*n - 11)*x^2 + 2*x^3)*v[i - 1] + ((12*n - 4)*x + (24*n - 12)*x^2 + (3*n - 9)*x^3 - x^4)*deriv(v[i - 1], x))/(-6*n^2 + 14*n - 8 + (3*n^2 + n - 4)*x - (n - 1)*x^2)); 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-152ea674385671f9
• 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-152ea674385671f9
• 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-152ea674385671f9
• 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-152ea674385671f9

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-152ea674385671f9

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

  • premise (sequence): Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
    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-81aa6611c032ec70 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.