A081582 → 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:  9,    2;
  3: 17,   34,     3;
  4: 25,  194,    75,      4;
  5: 33,  482,   723,    132,      5;
  6: 41,  898,  3483,   1796,    205,     6;
  7: 49, 1442,  9891,  13188,   3605,   294,     7;
  8: 57, 2114, 21483,  58884,  35805,  6342,   399,   8;
  9: 65, 2914, 39795, 179844, 224805, 79590, 10199, 520, 9;

The row polynomials satisfy -(8*n^2-23*n+15-(8*n^2-6*n-2)*x+(n-1)*x^2)*P(n,x) = -(8*n^2-15*n+7+(120*n^2-474*n+308)*x-(39*n-51)*x^2+2*x^3)*P(n-1,x) + ((8*n-7)*x+(240*n-211)*x^2+(8*n-37)*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[(-(8*n^2 - 15*n + 7 + (120*n^2 - 474*n + 308)*x - (39*n - 51)*x^2 + 2*x^3)*P[n - 1, x] + ((8*n - 7)*x + (240*n - 211)*x^2 + (8*n - 37)*x^3 - x^4)*D[P[n - 1, x], x])/(-8*n^2 + 23*n - 15 + (8*n^2 - 6*n - 2)*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] = (-(8*n^2 - 15*n + 7 + (120*n^2 - 474*n + 308)*x - (39*n - 51)*x^2 + 2*x^3)*v[i - 1] + ((8*n - 7)*x + (240*n - 211)*x^2 + (8*n - 37)*x^3 - x^4)*deriv(v[i - 1], x))/(-8*n^2 + 23*n - 15 + (8*n^2 - 6*n - 2)*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-cc281e0cede67e51
• 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-cc281e0cede67e51
• 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-cc281e0cede67e51
• 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-cc281e0cede67e51

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-cc281e0cede67e51

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

  • premise (sequence): Pascal-(1,7,1) array.
    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-a31f5c33917b15a7 root: theorem check

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

    • preserver_image reversal, shift 0 e-7d90ede2da8763e3

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,7,1) array.
        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-a31f5c33917b15a7 (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.