A165253 → (θ+1)²

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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;
  2: 1,   4;
  3: 1,  12,    9;
  4: 1,  24,   45,   16;
  5: 1,  40,  135,  112,    25;
  6: 1,  60,  315,  448,   225,    36;
  7: 1,  84,  630, 1344,  1125,   396,   49;
  8: 1, 112, 1134, 3360,  4125,  2376,  637,  64;
  9: 1, 144, 1890, 7392, 12375, 10296, 4459, 960, 81;

The row polynomials satisfy (72*n-108-(44*n^3-198*n^2+298*n-150)*x+(2*n^5-15*n^4+44*n^3-63*n^2+44*n-12)*x^2)*P(n,x) = (72*n-108-(44*n^3-198*n^2+154*n-6)*x+(2*n^5-15*n^4+6*n^3+51*n^2-56*n+12)*x^2+(n^5-5*n^4+8*n^3-4*n^2)*x^3)*P(n-1,x) + (144*x-(88*n^2-176*n+12)*x^2+(4*n^4-16*n^3-6*n^2+44*n-12)*x^3+(n^4-4*n^3+4*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[((72*n - 108 - (44*n^3 - 198*n^2 + 154*n - 6)*x + (2*n^5 - 15*n^4 + 6*n^3 + 51*n^2 - 56*n + 12)*x^2 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^3)*P[n - 1, x] + (144*x - (88*n^2 - 176*n + 12)*x^2 + (4*n^4 - 16*n^3 - 6*n^2 + 44*n - 12)*x^3 + (n^4 - 4*n^3 + 4*n^2)*x^4)*D[P[n - 1, x], x])/(72*n - 108 - (44*n^3 - 198*n^2 + 298*n - 150)*x + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*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] = ((72*n - 108 - (44*n^3 - 198*n^2 + 154*n - 6)*x + (2*n^5 - 15*n^4 + 6*n^3 + 51*n^2 - 56*n + 12)*x^2 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^3)*v[i - 1] + (144*x - (88*n^2 - 176*n + 12)*x^2 + (4*n^4 - 16*n^3 - 6*n^2 + 44*n - 12)*x^3 + (n^4 - 4*n^3 + 4*n^2)*x^4)*deriv(v[i - 1], x))/(72*n - 108 - (44*n^3 - 198*n^2 + 298*n - 150)*x + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*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-0a6146aa5a7bedf3
• 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-0a6146aa5a7bedf3
• 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-0a6146aa5a7bedf3
• 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-0a6146aa5a7bedf3, e-ff7f89be7a8bc37a
• 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-ff7f89be7a8bc37a
• 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-ff7f89be7a8bc37a
• 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-ff7f89be7a8bc37a

Proof chains

2 incoming proof steps. 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-0a6146aa5a7bedf3

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

  • premise (sequence): Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-2580795600db03bb root: theorem check

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

    • theorem_check three_term_recurrence e-365c9804d88a701d root: theorem check

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

• preserver_image euler_polynomial, shift 0 e-ff7f89be7a8bc37a

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

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-cec419fd822ca749 root: theorem check

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

    • theorem_check three_term_recurrence e-59d957f5dca11392 root: theorem check

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

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