A116412 → 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:   3,    2;
  2:   6,   12,     3;
  3:  12,   42,    27,     4;
  4:  24,  120,   135,    48,     5;
  5:  48,  312,   513,   312,    75,     6;
  6:  96,  768,  1674,  1488,   600,   108,     7;
  7: 192, 1824,  4968,  5892,  3450,  1026,   147,    8;
  8: 384, 4224, 13824, 20640, 16125,  6912,  1617,  192,   9;
  9: 768, 9600, 36720, 66336, 65415, 37314, 12495, 2400, 243, 10;

The row polynomials satisfy -(8*n^2-8*n+(6*n^3-16*n^2-6*n)*x-(3*n^3+8*n^2-3*n)*x^2+(n^2+n)*x^3)*P(n,x) = -(16*n^2-16*n+(12*n^3+16*n^2-76*n)*x+(12*n^3-28*n^2-60*n+20)*x^2-(13*n^2+7*n-10)*x^3+(2*n+2)*x^4)*P(n-1,x) + (16*n*x+(12*n^2+44*n-8)*x^2+(24*n^2+24*n-20)*x^3+(3*n^2-5*n-10)*x^4-(n+1)*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[(-(16*n^2 - 16*n + (12*n^3 + 16*n^2 - 76*n)*x + (12*n^3 - 28*n^2 - 60*n + 20)*x^2 - (13*n^2 + 7*n - 10)*x^3 + (2*n + 2)*x^4)*P[n - 1, x] + (16*n*x + (12*n^2 + 44*n - 8)*x^2 + (24*n^2 + 24*n - 20)*x^3 + (3*n^2 - 5*n - 10)*x^4 - (n + 1)*x^5)*D[P[n - 1, x], x])/(-8*n^2 + 8*n - (6*n^3 - 16*n^2 - 6*n)*x + (3*n^3 + 8*n^2 - 3*n)*x^2 - (n^2 + 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] = (-(16*n^2 - 16*n + (12*n^3 + 16*n^2 - 76*n)*x + (12*n^3 - 28*n^2 - 60*n + 20)*x^2 - (13*n^2 + 7*n - 10)*x^3 + (2*n + 2)*x^4)*v[i - 1] + (16*n*x + (12*n^2 + 44*n - 8)*x^2 + (24*n^2 + 24*n - 20)*x^3 + (3*n^2 - 5*n - 10)*x^4 - (n + 1)*x^5)*deriv(v[i - 1], x))/(-8*n^2 + 8*n - (6*n^3 - 16*n^2 - 6*n)*x + (3*n^3 + 8*n^2 - 3*n)*x^2 - (n^2 + 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-0cc329e6d69bfe0a
• 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-0cc329e6d69bfe0a
• 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-0cc329e6d69bfe0a
• 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-0cc329e6d69bfe0a

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-0cc329e6d69bfe0a

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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. [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.