A059473 → 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:    2;
  2:   10,     8;
  3:   32,    64,     24;
  4:   88,   296,    264,     64;
  5:  224,  1072,   1608,    896,    160;
  6:  544,  3392,   7320,   6784,   2720,    384;
  7: 1280,  9856,  28032,  37376,  24640,   7680,    896;
  8: 2944, 27008,  95808, 169024, 159680,  81024,  20608,  2048;
  9: 6656, 70912, 302208, 668288, 835360, 604416, 248192, 53248, 4608;

The row polynomials satisfy -(3*n^2-7*n+4-(3*n^2+n-4)*x+(2*n-2)*x^2)*P(n,x) = -(6*n^2-8*n+2+(12*n^2-50*n+10)*x-(26*n-22)*x^2+8*x^3)*P(n-1,x) + ((6*n-2)*x+(24*n-12)*x^2+(6*n-18)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 - 8*n + 2 + (12*n^2 - 50*n + 10)*x - (26*n - 22)*x^2 + 8*x^3)*P[n - 1, x] + ((6*n - 2)*x + (24*n - 12)*x^2 + (6*n - 18)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(-3*n^2 + 7*n - 4 + (3*n^2 + n - 4)*x - (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n^2 - 8*n + 2 + (12*n^2 - 50*n + 10)*x - (26*n - 22)*x^2 + 8*x^3)*v[i - 1] + ((6*n - 2)*x + (24*n - 12)*x^2 + (6*n - 18)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(-3*n^2 + 7*n - 4 + (3*n^2 + n - 4)*x - (2*n - 2)*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-4447b916f35935f8
• 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-4447b916f35935f8
• 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-4447b916f35935f8
• 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-4447b916f35935f8

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-4447b916f35935f8

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

  • premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
    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-0ee228386582da48 root: theorem check

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

    • preserver_image reversal, shift 0 e-c68c82cb1005adf9

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
        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-0ee228386582da48 (expanded above)

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.