A145597 → 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,    6;
  2:  6,   30,    18;
  3: 10,   90,   135,     40;
  4: 15,  210,   567,    420,     75;
  5: 21,  420,  1764,   2352,   1050,    126;
  6: 28,  756,  4536,   9408,   7560,   2268,    196;
  7: 36, 1260, 10206,  30240,  37800,  20412,   4410,    288;
  8: 45, 1980, 20790,  83160, 148500, 124740,  48510,   7920,   405;
  9: 55, 2970, 39204, 203280, 490050, 588060, 355740, 104544, 13365, 550;

The row polynomials satisfy (2*n^2+12*n+(n^4+9*n^3+18*n^2)*x)*P(n,x) = (2*n^2+16*n+24+(n^4+11*n^3+48*n^2+76*n+32)*x+(3*n^4+23*n^3+58*n^2+60*n+24)*x^2)*P(n-1,x) + ((4*n+8)*x+(2*n^3+13*n^2+20*n+4)*x^2-(2*n^3+13*n^2+24*n+12)*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^2 + 16*n + 24 + (n^4 + 11*n^3 + 48*n^2 + 76*n + 32)*x + (3*n^4 + 23*n^3 + 58*n^2 + 60*n + 24)*x^2)*P[n - 1, x] + ((4*n + 8)*x + (2*n^3 + 13*n^2 + 20*n + 4)*x^2 - (2*n^3 + 13*n^2 + 24*n + 12)*x^3)*D[P[n - 1, x], x])/(2*n^2 + 12*n + (n^4 + 9*n^3 + 18*n^2)*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^2 + 16*n + 24 + (n^4 + 11*n^3 + 48*n^2 + 76*n + 32)*x + (3*n^4 + 23*n^3 + 58*n^2 + 60*n + 24)*x^2)*v[i - 1] + ((4*n + 8)*x + (2*n^3 + 13*n^2 + 20*n + 4)*x^2 - (2*n^3 + 13*n^2 + 24*n + 12)*x^3)*deriv(v[i - 1], x))/(2*n^2 + 12*n + (n^4 + 9*n^3 + 18*n^2)*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-1630b1dee35dc71d
• 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-1630b1dee35dc71d
• 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-1630b1dee35dc71d
• 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-1630b1dee35dc71d

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-1630b1dee35dc71d

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

  • premise (sequence): Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-bcb2e3135d28c083 root: theorem check

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

    • preserver_image reversal, shift 0 e-e8333902099cfe55

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-bcb2e3135d28c083 (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