A249251 → 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:  4,   3;
  2:  7,  18,    6;
  3: 10,  51,   57,    12;
  4: 13, 102,  225,   156,    24;
  5: 16, 171,  591,   795,   396,     48;
  6: 19, 258, 1236,  2604,  2478,    960,     96;
  7: 22, 363, 2241,  6582,  9753,   7128,   2256,   192;
  8: 25, 486, 3687, 14052, 29025,  32808,  19392,  5184,   384;
  9: 28, 627, 5655, 26661, 71634, 112587, 102264, 50640, 11712, 768;

The row polynomials satisfy -(3*n^3+n^2-2*n-(6*n^3-6*n^2-3*n+9)*x-(12*n^2-22*n+10)*x^2-(8*n^2-12*n-4)*x^3-(8*n-8)*x^4)*P(n,x) = -(3*n^3+4*n^2+n+(12*n^3-21*n-9)*x+(12*n^2-20*n-10)*x^2+(16*n^2+12*n+4)*x^3+(16*n+8)*x^4)*P(n-1,x) + ((3*n^2+4*n+1)*x+(24*n^2+40*n+13)*x^2+(12*n^2+84*n+52)*x^3+(64*n+88)*x^4+(16*n+64)*x^5+16*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[(-(3*n^3 + 4*n^2 + n + (12*n^3 - 21*n - 9)*x + (12*n^2 - 20*n - 10)*x^2 + (16*n^2 + 12*n + 4)*x^3 + (16*n + 8)*x^4)*P[n - 1, x] + ((3*n^2 + 4*n + 1)*x + (24*n^2 + 40*n + 13)*x^2 + (12*n^2 + 84*n + 52)*x^3 + (64*n + 88)*x^4 + (16*n + 64)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(-3*n^3 - n^2 + 2*n + (6*n^3 - 6*n^2 - 3*n + 9)*x + (12*n^2 - 22*n + 10)*x^2 + (8*n^2 - 12*n - 4)*x^3 + (8*n - 8)*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] = (-(3*n^3 + 4*n^2 + n + (12*n^3 - 21*n - 9)*x + (12*n^2 - 20*n - 10)*x^2 + (16*n^2 + 12*n + 4)*x^3 + (16*n + 8)*x^4)*v[i - 1] + ((3*n^2 + 4*n + 1)*x + (24*n^2 + 40*n + 13)*x^2 + (12*n^2 + 84*n + 52)*x^3 + (64*n + 88)*x^4 + (16*n + 64)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(-3*n^3 - n^2 + 2*n + (6*n^3 - 6*n^2 - 3*n + 9)*x + (12*n^2 - 22*n + 10)*x^2 + (8*n^2 - 12*n - 4)*x^3 + (8*n - 8)*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-6221f70b9611bcd0
• 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-6221f70b9611bcd0
• 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-6221f70b9611bcd0
• 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-6221f70b9611bcd0

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-6221f70b9611bcd0

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

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    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-67b682ca6511d839 root: theorem check

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

    • preserver_image reversal, shift 0 e-c7b023a4533e2987

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
        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-ee40b91a84fdd5b7 root: theorem check

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

        • preserver_image reversal, shift 0 e-c36241f3b645d7b5

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
            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-67b682ca6511d839 (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.