A249248 → (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,  12;
  2: 1,  28,    27;
  3: 1,  48,   198,     48;
  4: 1,  72,   621,    640,      75;
  5: 1, 100,  1431,   3856,    1525,     108;
  6: 1, 132,  2790,  14128,   14550,    3060,     147;
  7: 1, 168,  4887,  39440,   84825,   41580,    5488,    192;
  8: 1, 208,  7938,  92768,  345125,  352296,   99568,   9088,   243;
  9: 1, 252, 12186, 193760, 1106325, 2070756, 1145032, 210816, 14175, 300;

The row polynomials satisfy (n+1)*P(n,x) = (9*n^3+9*n^2-18*n)*x^2*P(n-4,x) + (4*n^3+4*n^2-8*n)*x^3*d/dx P(n-4,x) - ((4*n^2+8*n)*x-(18*n^2+27*n)*x^2)*P(n-3,x) + ((4*n^2+8*n)*x^2+(8*n^2+12*n)*x^3)*d/dx P(n-3,x) - (n+2-(4*n^2+4*n-4)*x-(9*n+9)*x^2)*P(n-2,x) - ((4*n^2+4*n-4)*x^2-(4*n+4)*x^3)*d/dx P(n-2,x) + (2*n+3+(4*n+4)*x)*P(n-1,x) - (4*n+4)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 12*x, P(2,x) = 1 + 28*x + 27*x^2, P(3,x) = 1 + 48*x + 198*x^2 + 48*x^3.

P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[2, x_] := 1 + 28*x + 27*x^2;
P[3, x_] := 1 + 48*x + 198*x^2 + 48*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 + 9*n^2 - 18*n)*x^2*P[n - 4, x] + (4*n^3 + 4*n^2 - 8*n)*x^3*D[P[n - 4, x], x] - ((4*n^2 + 8*n)*x - (18*n^2 + 27*n)*x^2)*P[n - 3, x] + ((4*n^2 + 8*n)*x^2 + (8*n^2 + 12*n)*x^3)*D[P[n - 3, x], x] - (n + 2 - (4*n^2 + 4*n - 4)*x - (9*n + 9)*x^2)*P[n - 2, x] - ((4*n^2 + 4*n - 4)*x^2 - (4*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n + 3 + (4*n + 4)*x)*P[n - 1, x] - (4*n + 4)*x^2*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x; v[3] = 1 + 28*x + 27*x^2; v[4] = 1 + 48*x + 198*x^2 + 48*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 + 9*n^2 - 18*n)*x^2*v[i - 4] + (4*n^3 + 4*n^2 - 8*n)*x^3*deriv(v[i - 4], x) - ((4*n^2 + 8*n)*x - (18*n^2 + 27*n)*x^2)*v[i - 3] + ((4*n^2 + 8*n)*x^2 + (8*n^2 + 12*n)*x^3)*deriv(v[i - 3], x) - (n + 2 - (4*n^2 + 4*n - 4)*x - (9*n + 9)*x^2)*v[i - 2] - ((4*n^2 + 4*n - 4)*x^2 - (4*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n + 3 + (4*n + 4)*x)*v[i - 1] - (4*n + 4)*x^2*deriv(v[i - 1], x))/(n + 1)); 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-7a524eebea10f044
• 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-7a524eebea10f044
• 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-7a524eebea10f044
• 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-7a524eebea10f044

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-7a524eebea10f044

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

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    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-2dcae0cd92b4b84a 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.