Real-rooted atlas

A145596 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(4\,n + 8\,n x + 4\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 20\,x^{2} + 8\,x^{3} + 5\,n + 20\,n x + 25\,n x^{2} + 10\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
23141431
34327248156
4560225330165785
56100550132014305723978
671541155400470076006200220335
782242184101922548034944247527072104856

Show all 15 stored rows

Generated b-files: A259476 ?

Computed in your browser ?

Technical details
Node id
n-d37595541a708e8b · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-db69b92786c145d5

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  2,   4;
  2:  3,  14,   14;
  3:  4,  32,   72,    48;
  4:  5,  60,  225,   330,    165;
  5:  6, 100,  550,  1320,   1430,    572;
  6:  7, 154, 1155,  4004,   7007,   6006,   2002;
  7:  8, 224, 2184, 10192,  25480,  34944,  24752,   7072;
  8:  9, 312, 3822, 22932,  76440, 148512, 167076, 100776,  25194;
  9: 10, 420, 6300, 47040, 199920, 514080, 813960, 775200, 406980, 90440;
%F Formula
The row polynomials satisfy (n^2+4*n+(2*n^2+8*n)*x+(n^2+4*n)*x^2)*P(n,x) = (n^2+5*n+4+(4*n^2+20*n+16)*x+(5*n^2+25*n+20)*x^2+(2*n^2+10*n+8)*x^3)*P(n-1,x) + ((2*n+2)*x+(6*n+6)*x^2+(6*n+6)*x^3+(2*n+2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 4 + (4*n^2 + 20*n + 16)*x + (5*n^2 + 25*n + 20)*x^2 + (2*n^2 + 10*n + 8)*x^3)*P[n - 1, x] + ((2*n + 2)*x + (6*n + 6)*x^2 + (6*n + 6)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 4 + (4*n^2 + 20*n + 16)*x + (5*n^2 + 25*n + 20)*x^2 + (2*n^2 + 10*n + 8)*x^3)*v[i - 1] + ((2*n + 2)*x + (6*n + 6)*x^2 + (6*n + 6)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*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)

Generated with polytool c48629374597.

A259476: triangle, offset 4, from the row P0
%e Example
Triangle begins:
   4:  1;
   5:  2,   4;
   6:  3,  14,   14;
   7:  4,  32,   72,    48;
   8:  5,  60,  225,   330,    165;
   9:  6, 100,  550,  1320,   1430,    572;
  10:  7, 154, 1155,  4004,   7007,   6006,   2002;
  11:  8, 224, 2184, 10192,  25480,  34944,  24752,   7072;
  12:  9, 312, 3822, 22932,  76440, 148512, 167076, 100776,  25194;
  13: 10, 420, 6300, 47040, 199920, 514080, 813960, 775200, 406980, 90440;
%F Formula
The row polynomials satisfy (n^2-4*n+(2*n^2-8*n)*x+(n^2-4*n)*x^2)*P(n,x) = (n^2-3*n+(4*n^2-12*n)*x+(5*n^2-15*n)*x^2+(2*n^2-6*n)*x^3)*P(n-1,x) + ((2*n-6)*x+(6*n-18)*x^2+(6*n-18)*x^3+(2*n-6)*x^4)*d/dx P(n-1,x) for n >= 5, with P(4,x) = 1.
%t Mathematica
P[4, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 3*n + (4*n^2 - 12*n)*x + (5*n^2 - 15*n)*x^2 + (2*n^2 - 6*n)*x^3)*P[n - 1, x] + ((2*n - 6)*x + (6*n - 18)*x^2 + (6*n - 18)*x^3 + (2*n - 6)*x^4)*D[P[n - 1, x], x])/(n^2 - 4*n + (2*n^2 - 8*n)*x + (n^2 - 4*n)*x^2)]];
Table[PadRight[CoefficientList[P[n, x], x], n - 3], {n, 4, 13}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 3)); v[1] = 1; for(n = 5, N, my(i = n - 3); v[i] = ((n^2 - 3*n + (4*n^2 - 12*n)*x + (5*n^2 - 15*n)*x^2 + (2*n^2 - 6*n)*x^3)*v[i - 1] + ((2*n - 6)*x + (6*n - 18)*x^2 + (6*n - 18)*x^3 + (2*n - 6)*x^4)*deriv(v[i - 1], x))/(n^2 - 4*n + (2*n^2 - 8*n)*x + (n^2 - 4*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 3 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 3, v[i])))};
T(13)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-a1fbec8352aea886
  • 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-a1fbec8352aea886
  • 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-a1fbec8352aea886

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 (2)

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