A391533 → 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: 4,  8,   3;
  1: 4, 16,  15,    4;
  2: 4, 24,  39,   24,    5;
  3: 4, 32,  75,   76,   35,    6;
  4: 4, 40, 123,  176,  130,   48,    7;
  5: 4, 48, 183,  340,  350,  204,   63,    8;
  6: 4, 56, 255,  584,  775,  624,  301,   80,    9;
  7: 4, 64, 339,  924, 1505, 1554, 1029,  424,   99,  10;
  8: 4, 72, 435, 1376, 2660, 3360, 2842, 1600,  576, 120,  11;
  9: 4, 80, 543, 1956, 4380, 6552, 6762, 4848, 2376, 760, 143, 12;

The row polynomials satisfy (2+(2*n+3)*x+(n+2)*x^2)*P(n,x) = (2+(2*n+7)*x+(3*n+8)*x^2+(n+3)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 8*x + 3*x^2.

P[0, x_] := 4 + 8*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (2*n + 7)*x + (3*n + 8)*x^2 + (n + 3)*x^3)*P[n - 1, x])/(2 + (2*n + 3)*x + (n + 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 8*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (2*n + 7)*x + (3*n + 8)*x^2 + (n + 3)*x^3)*v[i - 1])/(2 + (2*n + 3)*x + (n + 2)*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-d15a8c8191481e5e
• 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-d15a8c8191481e5e
• 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-d15a8c8191481e5e
• 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-d15a8c8191481e5e

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-d15a8c8191481e5e

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

  • premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-5e5f630c1e2baa8a root: theorem check

      Parent-row provers of polytool 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.