A094816 → (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,      4;
   3: 1,     12,       9;
   4: 1,     32,      54,      16;
   5: 1,     96,     261,     160,      25;
   6: 1,    356,    1305,    1200,     375,      36;
   7: 1,   1660,    7326,    8720,    4000,     756,     49;
   8: 1,   9488,   47187,   66864,   39375,   10836,   1372,    64;
   9: 1,  64288,  347562,  557760,  391475,  138096,  25382,  2304,   81;
  10: 1, 502692, 2895210, 5102816, 4079775, 1719900, 405426, 53376, 3645, 100;

The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-11*n+15+(4*n-12)*x)*P(n-3,x) - (4*n-12)*x^2*d/dx P(n-3,x) - (n^2-2*n-1+(4*n-8)*x-9*x^2)*P(n-2,x) + ((4*n-8)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n-3+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 12*x + 9*x^2, P(4,x) = 1 + 32*x + 54*x^2 + 16*x^3.

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[3, x_] := 1 + 12*x + 9*x^2;
P[4, x_] := 1 + 32*x + 54*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 11*n + 15 + (4*n - 12)*x)*P[n - 3, x] - (4*n - 12)*x^2*D[P[n - 3, x], x] - (n^2 - 2*n - 1 + (4*n - 8)*x - 9*x^2)*P[n - 2, x] + ((4*n - 8)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n - 3 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 12*x + 9*x^2; v[4] = 1 + 32*x + 54*x^2 + 16*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 11*n + 15 + (4*n - 12)*x)*v[i - 3] - (4*n - 12)*x^2*deriv(v[i - 3], x) - (n^2 - 2*n - 1 + (4*n - 8)*x - 9*x^2)*v[i - 2] + ((4*n - 8)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n - 3 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], x)); 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-c2ada5d59e3b7485
• 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-c2ada5d59e3b7485
• 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-c2ada5d59e3b7485
• 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-c2ada5d59e3b7485

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-c2ada5d59e3b7485

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