A046716 → 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,  2;
   3: 1,  6,    3;
   4: 1, 12,   24,     4;
   5: 1, 20,   87,    96,      5;
   6: 1, 30,  225,   580,    445,      6;
   7: 1, 42,  480,  2180,   4070,   2490,       7;
   8: 1, 56,  903,  6300,  20895,  31458,   16604,       8;
   9: 1, 72, 1554, 15344,  78295, 209160,  270326,  128576,       9;
  10: 1, 90, 2502, 33096, 238875, 979146, 2232482, 2573520, 1131057, 10;

The row polynomials satisfy (2*n-4)*P(n,x) = -((6*n^2-24*n+18)*x^2+(2*n^3-24*n^2+82*n-84)*x^3)*P(n-3,x) - ((2*n^2-8*n+6)*x^3+(n^3-10*n^2+31*n-30)*x^4)*d/dx P(n-3,x) - (2*n-2-(4*n-8)*x+(6*n^2-24*n+24)*x^2)*P(n-2,x) + ((2*n-2)*x+(3*n^2-13*n+14)*x^2+(n^3-9*n^2+24*n-20)*x^3)*d/dx P(n-2,x) + (4*n-6+(4*n^2-16*n+16)*x)*P(n-1,x) - ((2*n-2)*x+(n^2-7*n+10)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 1 + 6*x + 3*x^2.

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 1 + 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-((6*n^2 - 24*n + 18)*x^2 + (2*n^3 - 24*n^2 + 82*n - 84)*x^3)*P[n - 3, x] - ((2*n^2 - 8*n + 6)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*D[P[n - 3, x], x] - (2*n - 2 - (4*n - 8)*x + (6*n^2 - 24*n + 24)*x^2)*P[n - 2, x] + ((2*n - 2)*x + (3*n^2 - 13*n + 14)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*D[P[n - 2, x], x] + (4*n - 6 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] - ((2*n - 2)*x + (n^2 - 7*n + 10)*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 6*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-((6*n^2 - 24*n + 18)*x^2 + (2*n^3 - 24*n^2 + 82*n - 84)*x^3)*v[i - 3] - ((2*n^2 - 8*n + 6)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*deriv(v[i - 3], x) - (2*n - 2 - (4*n - 8)*x + (6*n^2 - 24*n + 24)*x^2)*v[i - 2] + ((2*n - 2)*x + (3*n^2 - 13*n + 14)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*deriv(v[i - 2], x) + (4*n - 6 + (4*n^2 - 16*n + 16)*x)*v[i - 1] - ((2*n - 2)*x + (n^2 - 7*n + 10)*x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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-e1869a5dbf71f6ee
• 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-e1869a5dbf71f6ee
• 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-e1869a5dbf71f6ee
• 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-e1869a5dbf71f6ee

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-e1869a5dbf71f6ee

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

  • premise (sequence): Coefficients of a special case of Poisson-Charlier polynomials.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-3fcbce34648f287c root: theorem check

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

    • preserver_image reversal, shift 0 e-1f661e973532853d

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-503a12a637bb094e root: theorem check

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

        • preserver_image reversal, shift 0 e-10ee31afdeccb0f4

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Coefficients of a special case of Poisson-Charlier polynomials.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-3fcbce34648f287c (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