Real-rooted atlas

A059473 → 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
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(6 + 54\,x + 180\,x^{2} + 300\,x^{3} + 270\,x^{4} + 126\,x^{5} + 24\,x^{6} - 9\,n - 76\,n x - 252\,n x^{2} - 426\,n x^{3} - 391\,n x^{4} - 186\,n x^{5} - 36\,n x^{6} + 5\,n^{2} + 32\,n^{2} x + 90\,n^{2} x^{2} + 140\,n^{2} x^{3} + 125\,n^{2} x^{4} + 60\,n^{2} x^{5} + 12\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(2 + 40\,x + 218\,x^{2} + 560\,x^{3} + 790\,x^{4} + 632\,x^{5} + 270\,x^{6} + 48\,x^{7} - 2\,n - 24\,n x - 112\,n x^{2} - 272\,n x^{3} - 378\,n x^{4} - 304\,n x^{5} - 132\,n x^{6} - 24\,n x^{7}\right)\,P_{n-1}'(x) + \left(2\,x + 22\,x^{2} + 92\,x^{3} + 200\,x^{4} + 250\,x^{5} + 182\,x^{6} + 72\,x^{7} + 12\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
24181840
385612080264
4161525087123561744
532384175238083960158411520
6649285376159442559221144704876096
71282176152965766412608016019210976031360502656

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-22966acca1036f68 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ab8a6f908e9a31d4

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,     4;
  2:   4,    18,     18;
  3:   8,    56,    120,     80;
  4:  16,   152,    508,    712,     356;
  5:  32,   384,   1752,   3808,    3960,    1584;
  6:  64,   928,   5376,  15944,   25592,   21144,    7048;
  7: 128,  2176,  15296,  57664,  126080,  160192,  109760,   31360;
  8: 256,  4992,  41280, 189120,  525456,  907776,  953760,  558144,  139536;
  9: 512, 11264, 107136, 578304, 1953312, 4284864, 6111168, 5471232, 2793888, 620864;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (5*n^2-9*n+6+(32*n^2-76*n+54)*x+(90*n^2-252*n+180)*x^2+(140*n^2-426*n+300)*x^3+(125*n^2-391*n+270)*x^4+(60*n^2-186*n+126)*x^5+(12*n^2-36*n+24)*x^6)*P(n-1,x) - (2*n-2+(24*n-40)*x+(112*n-218)*x^2+(272*n-560)*x^3+(378*n-790)*x^4+(304*n-632)*x^5+(132*n-270)*x^6+(24*n-48)*x^7)*d/dx P(n-1,x) + (2*x+22*x^2+92*x^3+200*x^4+250*x^5+182*x^6+72*x^7+12*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[((5*n^2 - 9*n + 6 + (32*n^2 - 76*n + 54)*x + (90*n^2 - 252*n + 180)*x^2 + (140*n^2 - 426*n + 300)*x^3 + (125*n^2 - 391*n + 270)*x^4 + (60*n^2 - 186*n + 126)*x^5 + (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] - (2*n - 2 + (24*n - 40)*x + (112*n - 218)*x^2 + (272*n - 560)*x^3 + (378*n - 790)*x^4 + (304*n - 632)*x^5 + (132*n - 270)*x^6 + (24*n - 48)*x^7)*D[P[n - 1, x], x] + (2*x + 22*x^2 + 92*x^3 + 200*x^4 + 250*x^5 + 182*x^6 + 72*x^7 + 12*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] = ((5*n^2 - 9*n + 6 + (32*n^2 - 76*n + 54)*x + (90*n^2 - 252*n + 180)*x^2 + (140*n^2 - 426*n + 300)*x^3 + (125*n^2 - 391*n + 270)*x^4 + (60*n^2 - 186*n + 126)*x^5 + (12*n^2 - 36*n + 24)*x^6)*v[i - 1] - (2*n - 2 + (24*n - 40)*x + (112*n - 218)*x^2 + (272*n - 560)*x^3 + (378*n - 790)*x^4 + (304*n - 632)*x^5 + (132*n - 270)*x^6 + (24*n - 48)*x^7)*deriv(v[i - 1], x) + (2*x + 22*x^2 + 92*x^3 + 200*x^4 + 250*x^5 + 182*x^6 + 72*x^7 + 12*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-c70dee1c4800e8a5
  • 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-c70dee1c4800e8a5
  • 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-c70dee1c4800e8a5

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