Real-rooted atlas

A143683 → 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
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(18 + 162\,x + 540\,x^{2} + 900\,x^{3} + 810\,x^{4} + 378\,x^{5} + 72\,x^{6} - 27\,n - 233\,n x - 771\,n x^{2} - 1293\,n x^{3} - 1178\,n x^{4} - 558\,n x^{5} - 108\,n x^{6} + 10\,n^{2} + 76\,n^{2} x + 240\,n^{2} x^{2} + 400\,n^{2} x^{3} + 370\,n^{2} x^{4} + 180\,n^{2} x^{5} + 36\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 95\,x + 604\,x^{2} + 1630\,x^{3} + 2345\,x^{4} + 1891\,x^{5} + 810\,x^{6} + 144\,x^{7} - n - 42\,n x - 266\,n x^{2} - 736\,n x^{3} - 1089\,n x^{4} - 902\,n x^{5} - 396\,n x^{6} - 72\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 41\,x^{2} + 226\,x^{3} + 550\,x^{4} + 725\,x^{5} + 541\,x^{6} + 216\,x^{7} + 36\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21121225
31226040123
4132208352176769
5142456142416806724275
6152804425690088256275225129
71621252984832560511043808010880143787

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-25e57be690fee1ab · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0cedec6a1b8ae11f

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 ?

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.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1, 12,   12;
  3: 1, 22,   60,    40;
  4: 1, 32,  208,   352,    176;
  5: 1, 42,  456,  1424,   1680,    672;
  6: 1, 52,  804,  4256,   9008,   8256,    2752;
  7: 1, 62, 1252,  9848,  32560,  51104,   38080,   10880;
  8: 1, 72, 1800, 19200,  92736, 222336,  278400,  175104,  43776;
  9: 1, 82, 2448, 33312, 219936, 747072, 1392384, 1445376, 785664, 174592;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (10*n^2-27*n+18+(76*n^2-233*n+162)*x+(240*n^2-771*n+540)*x^2+(400*n^2-1293*n+900)*x^3+(370*n^2-1178*n+810)*x^4+(180*n^2-558*n+378)*x^5+(36*n^2-108*n+72)*x^6)*P(n-1,x) - (n-1+(42*n-95)*x+(266*n-604)*x^2+(736*n-1630)*x^3+(1089*n-2345)*x^4+(902*n-1891)*x^5+(396*n-810)*x^6+(72*n-144)*x^7)*d/dx P(n-1,x) + (x+41*x^2+226*x^3+550*x^4+725*x^5+541*x^6+216*x^7+36*x^8)*d^2/dx^2 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[((10*n^2 - 27*n + 18 + (76*n^2 - 233*n + 162)*x + (240*n^2 - 771*n + 540)*x^2 + (400*n^2 - 1293*n + 900)*x^3 + (370*n^2 - 1178*n + 810)*x^4 + (180*n^2 - 558*n + 378)*x^5 + (36*n^2 - 108*n + 72)*x^6)*P[n - 1, x] - (n - 1 + (42*n - 95)*x + (266*n - 604)*x^2 + (736*n - 1630)*x^3 + (1089*n - 2345)*x^4 + (902*n - 1891)*x^5 + (396*n - 810)*x^6 + (72*n - 144)*x^7)*D[P[n - 1, x], x] + (x + 41*x^2 + 226*x^3 + 550*x^4 + 725*x^5 + 541*x^6 + 216*x^7 + 36*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = ((10*n^2 - 27*n + 18 + (76*n^2 - 233*n + 162)*x + (240*n^2 - 771*n + 540)*x^2 + (400*n^2 - 1293*n + 900)*x^3 + (370*n^2 - 1178*n + 810)*x^4 + (180*n^2 - 558*n + 378)*x^5 + (36*n^2 - 108*n + 72)*x^6)*v[i - 1] - (n - 1 + (42*n - 95)*x + (266*n - 604)*x^2 + (736*n - 1630)*x^3 + (1089*n - 2345)*x^4 + (902*n - 1891)*x^5 + (396*n - 810)*x^6 + (72*n - 144)*x^7)*deriv(v[i - 1], x) + (x + 41*x^2 + 226*x^3 + 550*x^4 + 725*x^5 + 541*x^6 + 216*x^7 + 36*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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.

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-6f46dc63d04bd23c
  • 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-6f46dc63d04bd23c
  • 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-6f46dc63d04bd23c

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 (1)

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