type_b_eulerian → γ (scale 1/4) → 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: 1,    2;
  3: 1,   10;
  4: 1,   36,     15;
  5: 1,  116,    183;
  6: 1,  358,   1437,    244;
  7: 1, 1086,   9333,   5540;
  8: 1, 3272,  54810,  76112,   6925;
  9: 1, 9832, 303498, 825104, 252605;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n + 1 + (2*n^2 - 2*n - 4)*x - (n^3 - n^2 - 2*n)*x^2)*v[i - 2] + ((2*n + 2)*x - (2*n^2 + 4*n + 2)*x^2 + (2*n^2 + 2*n)*x^3)*deriv(v[i - 2], x) - (1 - (2*n^2 - 2*n)*x)*v[i - 1] + (2*n*x - 2*n*x^2)*deriv(v[i - 1], x))/(n)); 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-b88134e74c47445b
• 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-b88134e74c47445b
• 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-b88134e74c47445b

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-b88134e74c47445b

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1/4, shift 0 e-4c46966ead0b055f

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): type_b_eulerian
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-4675d8b985e3f46a 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.