A063007 → 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:
   1: 1;
   2: 1,   4;
   3: 1,  12,   18;
   4: 1,  24,   90,    80;
   5: 1,  40,  270,   560,    350;
   6: 1,  60,  630,  2240,   3150,    1512;
   7: 1,  84, 1260,  6720,  15750,   16632,    6468;
   8: 1, 112, 2268, 16800,  57750,   99792,   84084,   27456;
   9: 1, 144, 3780, 36960, 173250,  432432,  588588,  411840,  115830;
  10: 1, 180, 5940, 73920, 450450, 1513512, 2942940, 3294720, 1969110, 486200;

The row polynomials satisfy -(n-1-(n^3-4*n^2+5*n-2)*x)*P(n,x) = -(n-1-(n^3-4*n^2+n+2)*x-(2*n^3-6*n^2+4*n)*x^2)*P(n-1,x) - (2*x-(2*n^2-4*n-2)*x^2-(2*n^2-4*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 - (n^3 - 4*n^2 + n + 2)*x - (2*n^3 - 6*n^2 + 4*n)*x^2)*P[n - 1, x] - (2*x - (2*n^2 - 4*n - 2)*x^2 - (2*n^2 - 4*n)*x^3)*D[P[n - 1, x], x])/(-n + 1 + (n^3 - 4*n^2 + 5*n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - 1 - (n^3 - 4*n^2 + n + 2)*x - (2*n^3 - 6*n^2 + 4*n)*x^2)*v[i - 1] - (2*x - (2*n^2 - 4*n - 2)*x^2 - (2*n^2 - 4*n)*x^3)*deriv(v[i - 1], x))/(-n + 1 + (n^3 - 4*n^2 + 5*n - 2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-4000f98bcecb476b
• Coefficients of one sign. Every nonzero row, n >= 1: 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-4000f98bcecb476b
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-4000f98bcecb476b
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-4000f98bcecb476b

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-4000f98bcecb476b

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

  • premise (sequence): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-18de01688f1c88b0 root: theorem check

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

    • theorem_check three_term_recurrence e-93f30859d3148f4f root: theorem check

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

    • preserver_image reversal, shift 0 e-00d47151fcd2720e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-74c20febd3423e2e root: theorem check

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

        • preserver_image reversal, shift 0 e-5047c71cb920acb6

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-18de01688f1c88b0 (expanded above)
            • ↵ theorem_check three_term_recurrence e-93f30859d3148f4f (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