Real-rooted atlas

A121574 → 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
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(8 + 66\,x + 210\,x^{2} + 340\,x^{3} + 300\,x^{4} + 138\,x^{5} + 26\,x^{6} - 13\,n - 101\,n x - 314\,n x^{2} - 505\,n x^{3} - 446\,n x^{4} - 206\,n x^{5} - 39\,n x^{6} + 7\,n^{2} + 44\,n^{2} x + 119\,n^{2} x^{2} + 176\,n^{2} x^{3} + 149\,n^{2} x^{4} + 68\,n^{2} x^{5} + 13\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(4 + 60\,x + 292\,x^{2} + 700\,x^{3} + 940\,x^{4} + 724\,x^{5} + 300\,x^{6} + 52\,x^{7} - 4\,n - 40\,n x - 165\,n x^{2} - 366\,n x^{3} - 474\,n x^{4} - 360\,n x^{5} - 149\,n x^{6} - 26\,n x^{7}\right)\,P_{n-1}'(x) + \left(4\,x + 36\,x^{2} + 133\,x^{3} + 265\,x^{4} + 310\,x^{5} + 214\,x^{6} + 81\,x^{7} + 13\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
24131027
38406433145
416108265281109779
532272902146111583604185
6646562744600172454584118922483
712815367760214083485633522176483927120785
8256352020848694961426811848581476916657112970648891
9512793653920210832523018854023918334627385247228428373486025
101024176641354246077921768876348927747262764342321259087390689914148118727907
112048389123322881683712562588013019032212974242463582719754428104610303293724467280100611585
1240968499280000045173761704611245292372869100971213794791224849348711853841468310118641121543321540513739
138192184320189593611805952496603521490326723283422225378057056550415565860821333745076921618879084243998250972432903791865
14163843973124433920301847041400559364686098881166120944219284054531317898033381153347271686525515758857406238761101509218651683505015599986803

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Technical details
Node id
n-34f4b429d946016e · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e1118c67b24d5944

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 ?

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,    3;
  2:   4,   13,    10;
  3:   8,   40,    64,     33;
  4:  16,  108,   265,    281,    109;
  5:  32,  272,   902,   1461,   1158,    360;
  6:  64,  656,  2744,   6001,   7245,   4584,   1189;
  7: 128, 1536,  7760,  21408,  34856,  33522,  17648,   3927;
  8: 256, 3520, 20848,  69496, 142681, 184858, 147691,  66571,  12970;
  9: 512, 7936, 53920, 210832, 523018, 854023, 918334, 627385, 247228, 42837;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (7*n^2-13*n+8+(44*n^2-101*n+66)*x+(119*n^2-314*n+210)*x^2+(176*n^2-505*n+340)*x^3+(149*n^2-446*n+300)*x^4+(68*n^2-206*n+138)*x^5+(13*n^2-39*n+26)*x^6)*P(n-1,x) - (4*n-4+(40*n-60)*x+(165*n-292)*x^2+(366*n-700)*x^3+(474*n-940)*x^4+(360*n-724)*x^5+(149*n-300)*x^6+(26*n-52)*x^7)*d/dx P(n-1,x) + (4*x+36*x^2+133*x^3+265*x^4+310*x^5+214*x^6+81*x^7+13*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[((7*n^2 - 13*n + 8 + (44*n^2 - 101*n + 66)*x + (119*n^2 - 314*n + 210)*x^2 + (176*n^2 - 505*n + 340)*x^3 + (149*n^2 - 446*n + 300)*x^4 + (68*n^2 - 206*n + 138)*x^5 + (13*n^2 - 39*n + 26)*x^6)*P[n - 1, x] - (4*n - 4 + (40*n - 60)*x + (165*n - 292)*x^2 + (366*n - 700)*x^3 + (474*n - 940)*x^4 + (360*n - 724)*x^5 + (149*n - 300)*x^6 + (26*n - 52)*x^7)*D[P[n - 1, x], x] + (4*x + 36*x^2 + 133*x^3 + 265*x^4 + 310*x^5 + 214*x^6 + 81*x^7 + 13*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] = ((7*n^2 - 13*n + 8 + (44*n^2 - 101*n + 66)*x + (119*n^2 - 314*n + 210)*x^2 + (176*n^2 - 505*n + 340)*x^3 + (149*n^2 - 446*n + 300)*x^4 + (68*n^2 - 206*n + 138)*x^5 + (13*n^2 - 39*n + 26)*x^6)*v[i - 1] - (4*n - 4 + (40*n - 60)*x + (165*n - 292)*x^2 + (366*n - 700)*x^3 + (474*n - 940)*x^4 + (360*n - 724)*x^5 + (149*n - 300)*x^6 + (26*n - 52)*x^7)*deriv(v[i - 1], x) + (4*x + 36*x^2 + 133*x^3 + 265*x^4 + 310*x^5 + 214*x^6 + 81*x^7 + 13*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-bf773fe4ecdc3c15
  • 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-bf773fe4ecdc3c15
  • 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-bf773fe4ecdc3c15

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

No candidate equalities or related sequences.

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