A063007 → (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:
   1: 1;
   2: 1,   8;
   3: 1,  24,    54;
   4: 1,  48,   270,    320;
   5: 1,  80,   810,   2240,    1750;
   6: 1, 120,  1890,   8960,   15750,    9072;
   7: 1, 168,  3780,  26880,   78750,   99792,    45276;
   8: 1, 224,  6804,  67200,  288750,  598752,   588588,   219648;
   9: 1, 288, 11340, 147840,  866250, 2594592,  4120116,  3294720,  1042470;
  10: 1, 360, 17820, 295680, 2252250, 9081072, 20600580, 26357760, 17721990, 4862000;

The row polynomials satisfy (n-1-(4*n^3-16*n^2+21*n-9)*x+(n^5-7*n^4+19*n^3-25*n^2+16*n-4)*x^2)*P(n,x) = (n-1-(4*n^3-16*n^2+13*n-1)*x+(n^5-7*n^4+5*n^3+17*n^2-20*n+4)*x^2+(2*n^5-10*n^4+16*n^3-8*n^2)*x^3)*P(n-1,x) + (2*x-(8*n^2-16*n+2)*x^2+(2*n^4-8*n^3+16*n-4)*x^3+(2*n^4-8*n^3+8*n^2)*x^4)*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 - (4*n^3 - 16*n^2 + 13*n - 1)*x + (n^5 - 7*n^4 + 5*n^3 + 17*n^2 - 20*n + 4)*x^2 + (2*n^5 - 10*n^4 + 16*n^3 - 8*n^2)*x^3)*P[n - 1, x] + (2*x - (8*n^2 - 16*n + 2)*x^2 + (2*n^4 - 8*n^3 + 16*n - 4)*x^3 + (2*n^4 - 8*n^3 + 8*n^2)*x^4)*D[P[n - 1, x], x])/(n - 1 - (4*n^3 - 16*n^2 + 21*n - 9)*x + (n^5 - 7*n^4 + 19*n^3 - 25*n^2 + 16*n - 4)*x^2)]];
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 - (4*n^3 - 16*n^2 + 13*n - 1)*x + (n^5 - 7*n^4 + 5*n^3 + 17*n^2 - 20*n + 4)*x^2 + (2*n^5 - 10*n^4 + 16*n^3 - 8*n^2)*x^3)*v[i - 1] + (2*x - (8*n^2 - 16*n + 2)*x^2 + (2*n^4 - 8*n^3 + 16*n - 4)*x^3 + (2*n^4 - 8*n^3 + 8*n^2)*x^4)*deriv(v[i - 1], x))/(n - 1 - (4*n^3 - 16*n^2 + 21*n - 9)*x + (n^5 - 7*n^4 + 19*n^3 - 25*n^2 + 16*n - 4)*x^2)); 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-4e0fa4d6f32bf88e
• 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-4e0fa4d6f32bf88e
• 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-4e0fa4d6f32bf88e
• 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-4e0fa4d6f32bf88e

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-4e0fa4d6f32bf88e

  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