A059344 → (θ+1)²

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,   8;
  3: 1,  24;
  4: 1,  48,   108;
  5: 1,  80,   540;
  6: 1, 120,  1620,   1920;
  7: 1, 168,  3780,  13440;
  8: 1, 224,  7560,  53760,  42000;
  9: 1, 288, 13608, 161280, 378000;

The row polynomials satisfy (1+(3*n^2-n-2)*x+(n^4+2*n^3+n^2)*x^2)*P(n,x) = (1+(3*n^2+7*n-10)*x+(n^4+12*n^3+5*n^2-18*n+4)*x^2+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^3)*P(n-1,x) - (4*x^2+(12*n^2+20*n)*x^3+(4*n^4+24*n^3+52*n^2+48*n+16)*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[((1 + (3*n^2 + 7*n - 10)*x + (n^4 + 12*n^3 + 5*n^2 - 18*n + 4)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*P[n - 1, x] - (4*x^2 + (12*n^2 + 20*n)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*D[P[n - 1, x], x])/(1 + (3*n^2 - n - 2)*x + (n^4 + 2*n^3 + n^2)*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] = ((1 + (3*n^2 + 7*n - 10)*x + (n^4 + 12*n^3 + 5*n^2 - 18*n + 4)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*v[i - 1] - (4*x^2 + (12*n^2 + 20*n)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*deriv(v[i - 1], x))/(1 + (3*n^2 - n - 2)*x + (n^4 + 2*n^3 + n^2)*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-5ad1e40b769264f9
• 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-5ad1e40b769264f9
• 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-5ad1e40b769264f9

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-5ad1e40b769264f9

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

  • premise (sequence): Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers paper_theorem e-94e95c2ea0369f05 root: theorem check

      Parent-row provers of polytool polytool ?

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