A081580 → 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:  8,    9,     1;
  3: 14,   39,    16,     1;
  4: 20,  141,   118,    23,      1;
  5: 26,  315,   580,   245,     30,     1;
  6: 32,  561,  2050,  1655,    420,    37,     1;
  7: 38,  879,  5176,  7925,   3690,   643,    44,    1;
  8: 44, 1269, 10606, 27911,  22800,  7009,   914,   51,  1;
  9: 50, 1731, 18988, 75653, 104406, 53695, 11936, 1233, 58, 1;

The row polynomials satisfy -(6*n^3-16*n^2+10*n-(6*n^3-25*n^2+52*n)*x-(20*n^2-49*n)*x^2-(n^2+18*n)*x^3-n*x^4)*P(n,x) = -(6*n^3-10*n^2+4*n+(66*n^3-227*n^2+95*n)*x+(214*n^2-540*n+351)*x^2+(11*n^2+179*n-199)*x^3+(10*n-31)*x^4-x^5)*P(n-1,x) + ((6*n^2-4*n)*x+(132*n^2-67*n-10)*x^2+(6*n^2+459*n-199)*x^3+(43*n+453)*x^4+(n+43)*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[(-(6*n^3 - 10*n^2 + 4*n + (66*n^3 - 227*n^2 + 95*n)*x + (214*n^2 - 540*n + 351)*x^2 + (11*n^2 + 179*n - 199)*x^3 + (10*n - 31)*x^4 - x^5)*P[n - 1, x] + ((6*n^2 - 4*n)*x + (132*n^2 - 67*n - 10)*x^2 + (6*n^2 + 459*n - 199)*x^3 + (43*n + 453)*x^4 + (n + 43)*x^5 + x^6)*D[P[n - 1, x], x])/(-6*n^3 + 16*n^2 - 10*n + (6*n^3 - 25*n^2 + 52*n)*x + (20*n^2 - 49*n)*x^2 + (n^2 + 18*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] = (-(6*n^3 - 10*n^2 + 4*n + (66*n^3 - 227*n^2 + 95*n)*x + (214*n^2 - 540*n + 351)*x^2 + (11*n^2 + 179*n - 199)*x^3 + (10*n - 31)*x^4 - x^5)*v[i - 1] + ((6*n^2 - 4*n)*x + (132*n^2 - 67*n - 10)*x^2 + (6*n^2 + 459*n - 199)*x^3 + (43*n + 453)*x^4 + (n + 43)*x^5 + x^6)*deriv(v[i - 1], x))/(-6*n^3 + 16*n^2 - 10*n + (6*n^3 - 25*n^2 + 52*n)*x + (20*n^2 - 49*n)*x^2 + (n^2 + 18*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-5eeadacb5f413009
• 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-5eeadacb5f413009
• 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-5eeadacb5f413009
• 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-5eeadacb5f413009

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-5eeadacb5f413009

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

  • premise (sequence): Pascal-(1,5,1) 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-307703197d4a988b root: theorem check

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

    • preserver_image reversal, shift 0 e-bf7624d7e60440c3

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,5,1) 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-307703197d4a988b (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.