Real-rooted atlas

A114655 → reverse

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 + n\right)\,P_{n}(x) = \left(1 - 6\,x + n + 6\,n x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
122
2246
3212822
4224481690
524016016032394
6260400800480641806
72848402800336013441288558
82112156878401568012544358425641586

Show all 15 stored rows

Generated b-files: A114655 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-435b4d5bded8f11f · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-9e57d05d9d9a1ce7

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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.

The atlas's own rows: n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 2;
   2: 2,   4;
   3: 2,  12,    8;
   4: 2,  24,   48,    16;
   5: 2,  40,  160,   160,     32;
   6: 2,  60,  400,   800,    480,     64;
   7: 2,  84,  840,  2800,   3360,   1344,    128;
   8: 2, 112, 1568,  7840,  15680,  12544,   3584,    256;
   9: 2, 144, 2688, 18816,  56448,  75264,  43008,   9216,   512;
  10: 2, 180, 4320, 40320, 169344, 338688, 322560, 138240, 23040, 1024;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (n+1+(6*n-6)*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (6*n - 6)*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n + 1 + (6*n - 6)*x)*v[i - 1] + (2*x - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

A114655: reversed triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1:    2;
   2:    4,     2;
   3:    8,    12,      2;
   4:   16,    48,     24,      2;
   5:   32,   160,    160,     40,      2;
   6:   64,   480,    800,    400,     60,      2;
   7:  128,  1344,   3360,   2800,    840,     84,     2;
   8:  256,  3584,  12544,  15680,   7840,   1568,   112,    2;
   9:  512,  9216,  43008,  75264,  56448,  18816,  2688,  144,   2;
  10: 1024, 23040, 138240, 322560, 338688, 169344, 40320, 4320, 180, 2;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (n+1+(6*n-6)*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (6*n - 6)*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n + 1 + (6*n - 6)*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-b0d131c104e489dd
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-b0d131c104e489dd
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-b0d131c104e489dd

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

Same fingerprint (5)

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

References

  1. [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