A121574 → 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:   2,    2;
  2:   4,   10,     3;
  3:   8,   32,    24,     4;
  4:  16,   88,   111,    44,     5;
  5:  32,  224,   402,   268,    70,     6;
  6:  64,  544,  1272,  1220,   530,   102,     7;
  7: 128, 1280,  3696,  4672,  2920,   924,   140,    8;
  8: 256, 2944, 10128, 15968, 13205,  5988,  1477,  184,   9;
  9: 512, 6656, 26592, 50368, 52210, 31326, 11018, 2216, 234, 10;

The row polynomials satisfy -(8*n+(6*n^2-8*n)*x-(3*n^2+4*n)*x^2)*P(n,x) = -(16*n+(12*n^2+32*n-32)*x+(12*n^2-2*n-28)*x^2-(6*n+8)*x^3)*P(n-1,x) + (16*x+(12*n+48)*x^2+(24*n+36)*x^3+(3*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n + (12*n^2 + 32*n - 32)*x + (12*n^2 - 2*n - 28)*x^2 - (6*n + 8)*x^3)*P[n - 1, x] + (16*x + (12*n + 48)*x^2 + (24*n + 36)*x^3 + (3*n + 4)*x^4)*D[P[n - 1, x], x])/(-8*n - (6*n^2 - 8*n)*x + (3*n^2 + 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(16*n + (12*n^2 + 32*n - 32)*x + (12*n^2 - 2*n - 28)*x^2 - (6*n + 8)*x^3)*v[i - 1] + (16*x + (12*n + 48)*x^2 + (24*n + 36)*x^3 + (3*n + 4)*x^4)*deriv(v[i - 1], x))/(-8*n - (6*n^2 - 8*n)*x + (3*n^2 + 4*n)*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-1afe601b62cd3a17
• 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-1afe601b62cd3a17
• 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-1afe601b62cd3a17
• 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-1afe601b62cd3a17

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-1afe601b62cd3a17

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

  • premise (sequence): Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
    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-81aa6611c032ec70 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.