Real-rooted atlas

A026729 → basis A008277 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 7\,x + 14\,x^{2} + 11\,x^{3} + 3\,x^{4} - n x - 3\,n x^{2} - 3\,n x^{3} - n x^{4}\right)\,P_{n-2}(x) + \left(x + 4\,x^{2} + 6\,x^{3} + 4\,x^{4} + x^{5}\right)\,P_{n-2}'(x) + \left(1 + 5\,x + 6\,x^{2} + 2\,x^{3} - n x - 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
328717
4322432593
5559191238102595
68152736148413474564183
71339326777774112167957221932249
821102294633721175963847664919111640269277
934268633077169547459762717845650405318563653642417283
105571281153307516572603308533621167130865105602215946439064623182487
11891907340323832850611415115336589839599477216294796841168501153014792472719236286841
121445136014181121426910475020926238111812488001134663474530596513967341684458113145771165087912548200109
1323313894750250526192227739184968414970004603733423682631146101073148315495740476556292161829387158890311583966128965176307
1437737713217955440269375867203031535892000343232736763868455951634262802679073428092098671656271053406257433818576982554412851634863345874850039
156101025914647035791177260967104845067865571781288519474076944447185913872281485091618910132548339919026276950475629322236472337128135065808124391865421348844326047080089
Technical details
Node id
n-3ca36ae872badac5 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a6723c5bbce0b58d

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 -1)
  • 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1,    2;
   3:  2,    8,      7;
   4:  3,   22,     43,     25;
   5:  5,   59,    191,    238,     102;
   6:  8,  152,    736,   1484,    1347,     456;
   7: 13,  393,   2677,   7774,   11216,    7957,    2219;
   8: 21, 1022,   9463,  37211,   75963,   84766,   49191,   11640;
   9: 34, 2686,  33077, 169547,  459762,  717845,  650405,  318563,   65364;
  10: 55, 7128, 115330, 751657, 2603308, 5336211, 6713086, 5105602, 2159464, 390646;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1-(n-7)*x-(3*n-14)*x^2-(3*n-11)*x^3-(n-3)*x^4)*P(n-2,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-2,x) + (1-(n-5)*x-(2*n-6)*x^2-(n-2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (n - 7)*x - (3*n - 14)*x^2 - (3*n - 11)*x^3 - (n - 3)*x^4)*P[n - 2, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 2, x], x] + (1 - (n - 5)*x - (2*n - 6)*x^2 - (n - 2)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = ((1 - (n - 7)*x - (3*n - 14)*x^2 - (3*n - 11)*x^3 - (n - 3)*x^4)*v[i - 2] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 2], x) + (1 - (n - 5)*x - (2*n - 6)*x^2 - (n - 2)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741