A116412 → 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,     1;
  2:   12,     8,     1;
  3:   33,    39,    12,     1;
  4:   84,   150,    81,    16,     1;
  5:  204,   498,   405,   138,    20,     1;
  6:  480,  1500,  1674,   852,   210,    24,    1;
  7: 1104,  4224,  6075,  4233,  1545,   297,   28,   1;
  8: 2496, 11328, 20088, 18060,  8985,  2538,  399,  32,  1;
  9: 5568, 29280, 61992, 68916, 44178, 16929, 3885, 516, 36, 1;

The row polynomials satisfy -(6*n^3-20*n^2+18*n-4-(3*n^3-10*n^2+49*n-42)*x-(4*n^2-12*n-8)*x^2-(n^2+3*n+6)*x^3-(n+1)*x^4)*P(n,x) = -(12*n^3-28*n^2+16+(12*n^3-40*n^2-76*n+104)*x+(16*n^2-96*n+144)*x^2+(4*n^2-28)*x^3+(2*n-20)*x^4-2*x^5)*P(n-1,x) + ((12*n^2-4*n-8)*x+(24*n^2+16*n-48)*x^2+(3*n^2+51*n-42)*x^3+(14*n+44)*x^4+(n+14)*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[(-(12*n^3 - 28*n^2 + 16 + (12*n^3 - 40*n^2 - 76*n + 104)*x + (16*n^2 - 96*n + 144)*x^2 + (4*n^2 - 28)*x^3 + (2*n - 20)*x^4 - 2*x^5)*P[n - 1, x] + ((12*n^2 - 4*n - 8)*x + (24*n^2 + 16*n - 48)*x^2 + (3*n^2 + 51*n - 42)*x^3 + (14*n + 44)*x^4 + (n + 14)*x^5 + x^6)*D[P[n - 1, x], x])/(-6*n^3 + 20*n^2 - 18*n + 4 + (3*n^3 - 10*n^2 + 49*n - 42)*x + (4*n^2 - 12*n - 8)*x^2 + (n^2 + 3*n + 6)*x^3 + (n + 1)*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] = (-(12*n^3 - 28*n^2 + 16 + (12*n^3 - 40*n^2 - 76*n + 104)*x + (16*n^2 - 96*n + 144)*x^2 + (4*n^2 - 28)*x^3 + (2*n - 20)*x^4 - 2*x^5)*v[i - 1] + ((12*n^2 - 4*n - 8)*x + (24*n^2 + 16*n - 48)*x^2 + (3*n^2 + 51*n - 42)*x^3 + (14*n + 44)*x^4 + (n + 14)*x^5 + x^6)*deriv(v[i - 1], x))/(-6*n^3 + 20*n^2 - 18*n + 4 + (3*n^3 - 10*n^2 + 49*n - 42)*x + (4*n^2 - 12*n - 8)*x^2 + (n^2 + 3*n + 6)*x^3 + (n + 1)*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-7fe4c3f24349ff5b
• 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-7fe4c3f24349ff5b
• 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-7fe4c3f24349ff5b
• 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-7fe4c3f24349ff5b

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-7fe4c3f24349ff5b

  Laguerre’s theorem on differential operators [SP14] 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 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 (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.