A274496 → 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;
  3:  4,   8;
  4:  4,  16,  12;
  5:  8,  32,  24;
  6:  8,  48,  72,  32;
  7: 16,  96, 144,  64;
  8: 16, 128, 288, 256,  80;
  9: 32, 256, 576, 512, 160;

The row polynomials satisfy (8+(4*n+6)*x+(n^2+n)*x^2)*P(n,x) = -(8*n-24+(6*n^2-28*n+2)*x+(n^3-10*n^2-7*n+12)*x^2-(n^3+2*n^2-n-2)*x^3)*P(n-1,x) + (8+(8*n-4)*x+(2*n^2-2*n-28)*x^2-(16*n+20)*x^3-(2*n^2+6*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[(-(8*n - 24 + (6*n^2 - 28*n + 2)*x + (n^3 - 10*n^2 - 7*n + 12)*x^2 - (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] + (8 + (8*n - 4)*x + (2*n^2 - 2*n - 28)*x^2 - (16*n + 20)*x^3 - (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(8 + (4*n + 6)*x + (n^2 + 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] = (-(8*n - 24 + (6*n^2 - 28*n + 2)*x + (n^3 - 10*n^2 - 7*n + 12)*x^2 - (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] + (8 + (8*n - 4)*x + (2*n^2 - 2*n - 28)*x^2 - (16*n + 20)*x^3 - (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(8 + (4*n + 6)*x + (n^2 + 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-aaacb3fb62ab0797
• 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-aaacb3fb62ab0797
• 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-aaacb3fb62ab0797

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-aaacb3fb62ab0797

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-ec4171dcd40223e2 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

• A152198 → D+1
• A274496 → D+1
• A152198 → θ+1

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.