A227450 → 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: 2;
  1: 2,    8;
  2: 2,   24,     24;
  3: 2,   56,    144,      64;
  4: 2,  120,    600,     640,     160;
  5: 2,  248,   2160,    4160,    2400,     384;
  6: 2,  504,   7224,   22400,   22400,    8064,     896;
  7: 2, 1016,  23184,  108864,  168000,  102144,   25088,    2048;
  8: 2, 2040,  72600,  497280, 1112160, 1016064,  413952,   73728,   4608;
  9: 2, 4088, 223920, 2182720, 6804000, 8765568, 5268480, 1536000, 207360, 10240;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -4*x^2*v[i - 2] - 2*x^2*deriv(v[i - 2], x) + (1 + 4*x)*v[i - 1] + x*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 - 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-36c887a5cdb807b7
• 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-36c887a5cdb807b7
• 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-36c887a5cdb807b7
• 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-36c887a5cdb807b7

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-36c887a5cdb807b7

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

  • premise (sequence): Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-74361f30d2bc482d root: theorem check

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

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.