A143683 → 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:  1;
  1:  2,    1;
  2: 11,   12,     1;
  3: 20,   57,    22,      1;
  4: 29,  264,   202,     32,      1;
  5: 38,  633,  1192,    446,     42,      1;
  6: 47, 1164,  5179,   3776,    789,     52,     1;
  7: 56, 1857, 14350,  22415,   8988,   1231,    62,    1;
  8: 65, 2712, 30892,  95000,  70566,  17800,  1772,   72,  1;
  9: 74, 3729, 56992, 286412, 402036, 177334, 31184, 2412, 82, 1;

The row polynomials satisfy -(9*n^3-25*n^2+16*n-(9*n^3-40*n^2+82*n)*x-(32*n^2-79*n)*x^2-(n^2+30*n)*x^3-n*x^4)*P(n,x) = -(9*n^3-16*n^2+7*n+(153*n^3-554*n^2+299*n)*x+(535*n^2-1341*n+849)*x^2+(17*n^2+479*n-511)*x^3+(16*n-49)*x^4-x^5)*P(n-1,x) + ((9*n^2-7*n)*x+(306*n^2-205*n-16)*x^2+(9*n^2+1116*n-511)*x^3+(67*n+1107)*x^4+(n+67)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^3 - 16*n^2 + 7*n + (153*n^3 - 554*n^2 + 299*n)*x + (535*n^2 - 1341*n + 849)*x^2 + (17*n^2 + 479*n - 511)*x^3 + (16*n - 49)*x^4 - x^5)*P[n - 1, x] + ((9*n^2 - 7*n)*x + (306*n^2 - 205*n - 16)*x^2 + (9*n^2 + 1116*n - 511)*x^3 + (67*n + 1107)*x^4 + (n + 67)*x^5 + x^6)*D[P[n - 1, x], x])/(-9*n^3 + 25*n^2 - 16*n + (9*n^3 - 40*n^2 + 82*n)*x + (32*n^2 - 79*n)*x^2 + (n^2 + 30*n)*x^3 + n*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(9*n^3 - 16*n^2 + 7*n + (153*n^3 - 554*n^2 + 299*n)*x + (535*n^2 - 1341*n + 849)*x^2 + (17*n^2 + 479*n - 511)*x^3 + (16*n - 49)*x^4 - x^5)*v[i - 1] + ((9*n^2 - 7*n)*x + (306*n^2 - 205*n - 16)*x^2 + (9*n^2 + 1116*n - 511)*x^3 + (67*n + 1107)*x^4 + (n + 67)*x^5 + x^6)*deriv(v[i - 1], x))/(-9*n^3 + 25*n^2 - 16*n + (9*n^3 - 40*n^2 + 82*n)*x + (32*n^2 - 79*n)*x^2 + (n^2 + 30*n)*x^3 + n*x^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-34544b66b04cbaa0
• 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-34544b66b04cbaa0
• 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-34544b66b04cbaa0
• 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-34544b66b04cbaa0

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-34544b66b04cbaa0

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

  • premise (sequence): Pascal-(1,8,1) 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-a922c934805482be root: theorem check

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

    • preserver_image reversal, shift 0 e-e737400fb7cbaf29

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,8,1) 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-a922c934805482be (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.