A144438 → 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: 1,    2;
  2: 1,   10,      3;
  3: 1,   28,     42,       4;
  4: 1,   66,    267,     132,       5;
  5: 1,  144,   1239,    1652,     360,        6;
  6: 1,  302,   4896,   13572,    8160,      906,       7;
  7: 1,  620,  17622,   88580,  110725,    35244,    2170,      8;
  8: 1, 1258,  59829,  501824, 1124215,   752736,  139601,   5032,     9;
  9: 1, 2536, 195513, 2589196, 9495480, 11394576, 4531093, 521368, 11412, 10;

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*n-1)*x^2*P(n-3,x) - (x^2-x^3)*d/dx P(n-3,x) + (2*x-(n^2-1)*x^2)*P(n-2,x) - ((n+1)*x^2-(n+1)*x^3)*d/dx P(n-2,x) + (1+2*n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 10*x + 3*x^2, P(3,x) = 1 + 28*x + 42*x^2 + 4*x^3.

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 10*x + 3*x^2;
P[3, x_] := 1 + 28*x + 42*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*n - 1)*x^2*P[n - 3, x] - (x^2 - x^3)*D[P[n - 3, x], x] + (2*x - (n^2 - 1)*x^2)*P[n - 2, x] - ((n + 1)*x^2 - (n + 1)*x^3)*D[P[n - 2, x], x] + (1 + 2*n*x)*P[n - 1, x] + (x - x^2)*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] = 1; v[2] = 1 + 2*x; v[3] = 1 + 10*x + 3*x^2; v[4] = 1 + 28*x + 42*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*n - 1)*x^2*v[i - 3] - (x^2 - x^3)*deriv(v[i - 3], x) + (2*x - (n^2 - 1)*x^2)*v[i - 2] - ((n + 1)*x^2 - (n + 1)*x^3)*deriv(v[i - 2], x) + (1 + 2*n*x)*v[i - 1] + (x - 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 - 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-39092463700c66ac
• 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-39092463700c66ac
• 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-39092463700c66ac
• 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-39092463700c66ac

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-39092463700c66ac

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

  • premise (sequence): Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-d401dbe9093b268d root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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.