A091044 → (θ+1)²

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,    8;
  2:  3,   40,    27;
  3:  4,  112,   252,     64;
  4:  5,  240,  1134,    960,     125;
  5:  6,  440,  3564,   6336,    2750,     216;
  6:  7,  728,  9009,  27456,   25025,    6552,     343;
  7:  8, 1120, 19656,  91520,  143000,   78624,   13720,    512;
  8:  9, 1632, 38556, 254592,  607750,  572832,  209916,  26112,   729;
  9: 10, 2280, 69768, 620160, 2099500, 3023280, 1899240, 496128, 46170, 1000;

The row polynomials satisfy (n-(10*n^3+n^2-n)*x+(4*n^5+4*n^4+n^3)*x^2)*P(n,x) = (n+1-(10*n^3+11*n^2-12*n+3)*x+(4*n^5+8*n^4-49*n^3-56*n^2+28*n+3)*x^2+(12*n^5+40*n^4+31*n^3-5*n^2-5*n-1)*x^3)*P(n-1,x) + (2*x-(20*n^2+32*n+6)*x^2+(8*n^4+32*n^3+58*n^2+46*n+6)*x^3-(8*n^4+32*n^3+38*n^2+14*n+2)*x^4)*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 + 1 - (10*n^3 + 11*n^2 - 12*n + 3)*x + (4*n^5 + 8*n^4 - 49*n^3 - 56*n^2 + 28*n + 3)*x^2 + (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^3)*P[n - 1, x] + (2*x - (20*n^2 + 32*n + 6)*x^2 + (8*n^4 + 32*n^3 + 58*n^2 + 46*n + 6)*x^3 - (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^4)*D[P[n - 1, x], x])/(n - (10*n^3 + n^2 - n)*x + (4*n^5 + 4*n^4 + n^3)*x^2)]];
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 + 1 - (10*n^3 + 11*n^2 - 12*n + 3)*x + (4*n^5 + 8*n^4 - 49*n^3 - 56*n^2 + 28*n + 3)*x^2 + (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^3)*v[i - 1] + (2*x - (20*n^2 + 32*n + 6)*x^2 + (8*n^4 + 32*n^3 + 58*n^2 + 46*n + 6)*x^3 - (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^4)*deriv(v[i - 1], x))/(n - (10*n^3 + n^2 - n)*x + (4*n^5 + 4*n^4 + n^3)*x^2)); 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-988bd1603bc5c7fc
• 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-988bd1603bc5c7fc
• 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-988bd1603bc5c7fc
• 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-988bd1603bc5c7fc

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-988bd1603bc5c7fc

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

  • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • preserver_image reversal, shift 0 e-75f779345b39f910

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023