A081581 → 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:  9,   10,     1;
  3: 16,   45,    18,      1;
  4: 23,  178,   144,     26,      1;
  5: 30,  409,   760,    306,     34,     1;
  6: 37,  738,  2895,   2252,    531,    42,     1;
  7: 44, 1165,  7578,  11695,   5144,   819,    50,    1;
  8: 51, 1690, 15838,  44070,  34796,  9926,  1170,   58,  1;
  9: 58, 2313, 28704, 124416, 172836, 83958, 17088, 1584, 66, 1;

The row polynomials satisfy -(7*n^3-19*n^2+12*n-(7*n^3-30*n^2+62*n)*x-(24*n^2-59*n)*x^2-(n^2+22*n)*x^3-n*x^4)*P(n,x) = -(7*n^3-12*n^2+5*n+(91*n^3-320*n^2+151*n)*x+(305*n^2-767*n+493)*x^2+(13*n^2+263*n-287)*x^3+(12*n-37)*x^4-x^5)*P(n-1,x) + ((7*n^2-5*n)*x+(182*n^2-105*n-12)*x^2+(7*n^2+646*n-287)*x^3+(51*n+639)*x^4+(n+51)*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[(-(7*n^3 - 12*n^2 + 5*n + (91*n^3 - 320*n^2 + 151*n)*x + (305*n^2 - 767*n + 493)*x^2 + (13*n^2 + 263*n - 287)*x^3 + (12*n - 37)*x^4 - x^5)*P[n - 1, x] + ((7*n^2 - 5*n)*x + (182*n^2 - 105*n - 12)*x^2 + (7*n^2 + 646*n - 287)*x^3 + (51*n + 639)*x^4 + (n + 51)*x^5 + x^6)*D[P[n - 1, x], x])/(-7*n^3 + 19*n^2 - 12*n + (7*n^3 - 30*n^2 + 62*n)*x + (24*n^2 - 59*n)*x^2 + (n^2 + 22*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] = (-(7*n^3 - 12*n^2 + 5*n + (91*n^3 - 320*n^2 + 151*n)*x + (305*n^2 - 767*n + 493)*x^2 + (13*n^2 + 263*n - 287)*x^3 + (12*n - 37)*x^4 - x^5)*v[i - 1] + ((7*n^2 - 5*n)*x + (182*n^2 - 105*n - 12)*x^2 + (7*n^2 + 646*n - 287)*x^3 + (51*n + 639)*x^4 + (n + 51)*x^5 + x^6)*deriv(v[i - 1], x))/(-7*n^3 + 19*n^2 - 12*n + (7*n^3 - 30*n^2 + 62*n)*x + (24*n^2 - 59*n)*x^2 + (n^2 + 22*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-e4ee601767287219
• 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-e4ee601767287219
• 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-e4ee601767287219
• 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-e4ee601767287219

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-e4ee601767287219

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

  • premise (sequence): Pascal-(1,6,1) 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-dfce858c9d52dee3 root: theorem check

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

    • preserver_image reversal, shift 0 e-5c5351a0cb2ba1df

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,6,1) 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-dfce858c9d52dee3 (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.