A249251 → θ+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: 1,  6;
  2: 1, 12,  18;
  3: 1, 18,  63,   48;
  4: 1, 24, 135,  240,   120;
  5: 1, 30, 234,  684,   780,   288;
  6: 1, 36, 360, 1488,  2790,  2304,    672;
  7: 1, 42, 513, 2760,  7365,  9936,   6384,  1536;
  8: 1, 48, 693, 4608, 16125, 30960,  32256, 16896,  3456;
  9: 1, 54, 900, 7140, 31095, 78498, 116088, 97920, 43200, 7680;

The row polynomials satisfy -(n^2+3*n+2+(3*n^3+6*n^2+5*n+10)*x-(6*n^3+12*n^2+2*n-20)*x^2-(8*n^2-8*n)*x^3)*P(n,x) = -(n^2+3*n+2+(3*n^3+18*n^2+29*n+10)*x+(12*n^3+48*n^2+64*n+20)*x^2+(4*n^2-4*n)*x^3)*P(n-1,x) + ((n+2)*x+(3*n^2+19*n+28)*x^2+(24*n^2+96*n+108)*x^3+(12*n^2+76*n+80)*x^4+(16*n+16)*x^5)*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[(-(n^2 + 3*n + 2 + (3*n^3 + 18*n^2 + 29*n + 10)*x + (12*n^3 + 48*n^2 + 64*n + 20)*x^2 + (4*n^2 - 4*n)*x^3)*P[n - 1, x] + ((n + 2)*x + (3*n^2 + 19*n + 28)*x^2 + (24*n^2 + 96*n + 108)*x^3 + (12*n^2 + 76*n + 80)*x^4 + (16*n + 16)*x^5)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 - (3*n^3 + 6*n^2 + 5*n + 10)*x + (6*n^3 + 12*n^2 + 2*n - 20)*x^2 + (8*n^2 - 8*n)*x^3)]];
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] = (-(n^2 + 3*n + 2 + (3*n^3 + 18*n^2 + 29*n + 10)*x + (12*n^3 + 48*n^2 + 64*n + 20)*x^2 + (4*n^2 - 4*n)*x^3)*v[i - 1] + ((n + 2)*x + (3*n^2 + 19*n + 28)*x^2 + (24*n^2 + 96*n + 108)*x^3 + (12*n^2 + 76*n + 80)*x^4 + (16*n + 16)*x^5)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 - (3*n^3 + 6*n^2 + 5*n + 10)*x + (6*n^3 + 12*n^2 + 2*n - 20)*x^2 + (8*n^2 - 8*n)*x^3)); 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-771904e9efc80f0d
• 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-771904e9efc80f0d
• 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-771904e9efc80f0d
• 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-771904e9efc80f0d

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 euler_polynomial, shift 0 e-771904e9efc80f0d

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.