A103327 → 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:  5,   20,     3;
  3:  7,   70,    63,      4;
  4:  9,  168,   378,    144,      5;
  5: 11,  330,  1386,   1320,    275,      6;
  6: 13,  572,  3861,   6864,   3575,    468,      7;
  7: 15,  910,  9009,  25740,  25025,   8190,    735,     8;
  8: 17, 1360, 18564,  77792, 121550,  74256,  16660,  1088,    9;
  9: 19, 1938, 34884, 201552, 461890, 453492, 189924, 31008, 1539, 10;

The row polynomials satisfy -(2*n-1-(4*n^3-4*n^2+n)*x)*P(n,x) = -(2*n+1-(4*n^3-13*n+10)*x-(12*n^3-13*n+3)*x^2)*P(n-1,x) - (4*x-(8*n^2+8*n+2)*x^2+(8*n^2+8*n-2)*x^3)*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[(-(2*n + 1 - (4*n^3 - 13*n + 10)*x - (12*n^3 - 13*n + 3)*x^2)*P[n - 1, x] - (4*x - (8*n^2 + 8*n + 2)*x^2 + (8*n^2 + 8*n - 2)*x^3)*D[P[n - 1, x], x])/(-2*n + 1 + (4*n^3 - 4*n^2 + n)*x)]];
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] = (-(2*n + 1 - (4*n^3 - 13*n + 10)*x - (12*n^3 - 13*n + 3)*x^2)*v[i - 1] - (4*x - (8*n^2 + 8*n + 2)*x^2 + (8*n^2 + 8*n - 2)*x^3)*deriv(v[i - 1], x))/(-2*n + 1 + (4*n^3 - 4*n^2 + n)*x)); 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-5eacdf1805a7ca9d
• 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-5eacdf1805a7ca9d
• 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-5eacdf1805a7ca9d
• 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-5eacdf1805a7ca9d

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-5eacdf1805a7ca9d

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

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-86d927bb13c60f05 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