Real-rooted atlas

A081579 → 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
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(10 + 90\,x + 300\,x^{2} + 500\,x^{3} + 450\,x^{4} + 210\,x^{5} + 40\,x^{6} - 15\,n - 129\,n x - 427\,n x^{2} - 717\,n x^{3} - 654\,n x^{4} - 310\,n x^{5} - 60\,n x^{6} + 6\,n^{2} + 44\,n^{2} x + 136\,n^{2} x^{2} + 224\,n^{2} x^{3} + 206\,n^{2} x^{4} + 100\,n^{2} x^{5} + 20\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 55\,x + 340\,x^{2} + 910\,x^{3} + 1305\,x^{4} + 1051\,x^{5} + 450\,x^{6} + 80\,x^{7} - n - 26\,n x - 154\,n x^{2} - 416\,n x^{3} - 609\,n x^{4} - 502\,n x^{5} - 220\,n x^{6} - 40\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 25\,x^{2} + 130\,x^{3} + 310\,x^{4} + 405\,x^{5} + 301\,x^{6} + 120\,x^{7} + 20\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
218817
3114362475
412010016080361
51262005606402561683
61323361440280024968327937
713850830168720128969408268837275

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-0821f66594bc073c · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-076e7110b92525fe

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: 1,  2;
  2: 1,  8,   8;
  3: 1, 14,  36,   24;
  4: 1, 20, 100,  160,    80;
  5: 1, 26, 200,  560,   640,    256;
  6: 1, 32, 336, 1440,  2800,   2496,    832;
  7: 1, 38, 508, 3016,  8720,  12896,   9408,   2688;
  8: 1, 44, 716, 5504, 21856,  47296,  56384,  34816,   8704;
  9: 1, 50, 960, 9120, 46960, 137952, 237440, 236800, 126720, 28160;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (6*n^2-15*n+10+(44*n^2-129*n+90)*x+(136*n^2-427*n+300)*x^2+(224*n^2-717*n+500)*x^3+(206*n^2-654*n+450)*x^4+(100*n^2-310*n+210)*x^5+(20*n^2-60*n+40)*x^6)*P(n-1,x) - (n-1+(26*n-55)*x+(154*n-340)*x^2+(416*n-910)*x^3+(609*n-1305)*x^4+(502*n-1051)*x^5+(220*n-450)*x^6+(40*n-80)*x^7)*d/dx P(n-1,x) + (x+25*x^2+130*x^3+310*x^4+405*x^5+301*x^6+120*x^7+20*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[((6*n^2 - 15*n + 10 + (44*n^2 - 129*n + 90)*x + (136*n^2 - 427*n + 300)*x^2 + (224*n^2 - 717*n + 500)*x^3 + (206*n^2 - 654*n + 450)*x^4 + (100*n^2 - 310*n + 210)*x^5 + (20*n^2 - 60*n + 40)*x^6)*P[n - 1, x] - (n - 1 + (26*n - 55)*x + (154*n - 340)*x^2 + (416*n - 910)*x^3 + (609*n - 1305)*x^4 + (502*n - 1051)*x^5 + (220*n - 450)*x^6 + (40*n - 80)*x^7)*D[P[n - 1, x], x] + (x + 25*x^2 + 130*x^3 + 310*x^4 + 405*x^5 + 301*x^6 + 120*x^7 + 20*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] = ((6*n^2 - 15*n + 10 + (44*n^2 - 129*n + 90)*x + (136*n^2 - 427*n + 300)*x^2 + (224*n^2 - 717*n + 500)*x^3 + (206*n^2 - 654*n + 450)*x^4 + (100*n^2 - 310*n + 210)*x^5 + (20*n^2 - 60*n + 40)*x^6)*v[i - 1] - (n - 1 + (26*n - 55)*x + (154*n - 340)*x^2 + (416*n - 910)*x^3 + (609*n - 1305)*x^4 + (502*n - 1051)*x^5 + (220*n - 450)*x^6 + (40*n - 80)*x^7)*deriv(v[i - 1], x) + (x + 25*x^2 + 130*x^3 + 310*x^4 + 405*x^5 + 301*x^6 + 120*x^7 + 20*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-1bbdf11925fb622c
  • 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-1bbdf11925fb622c
  • 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-1bbdf11925fb622c

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