Real-rooted atlas

A113953 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(19 + 90\,x + 156\,x^{2} + 118\,x^{3} + 33\,x^{4} - 17\,n - 80\,n x - 142\,n x^{2} - 112\,n x^{3} - 33\,n x^{4}\right)\,P_{n-1}(x) + \left(9 + 65\,x + 174\,x^{2} + 222\,x^{3} + 137\,x^{4} + 33\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
23141532
35378763192
4111043585282791280
52125111702646288912158192
6436103537107001775715282531953248
7851409984137469838231099717821923247344064
81713228262801203483385665982606475683921481016552228224
93417239674283613081226054272929839814923666924193338944444714417920
106831608616855510338964107574110345962028199825165800201600639410742194327193323264
111365353414116272845035129564024079219490539478141573366152748585108248481453314078496495603979776
12273177072987546759266438989153140794560366402268691986960940807197897591024570209706216501336371491113909091328
135461166867233293419750434112909991460067833137394086830434094844998089043600793845351361249502956265586102661352116242630325300041728
1410923359242544158950297364316806187143785613448472403011232519151223744241177344757669663712593109528744866900151229774254838467770710173575163745628160
Technical details
Node id
n-6056f51cdc205dcf · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d49be7aa1d7d4cfc

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   1,    3;
  2:   3,   14,    15;
  3:   5,   37,    87,     63;
  4:  11,  104,   358,    528,     279;
  5:  21,  251,  1170,   2646,    2889,    1215;
  6:  43,  610,  3537,  10700,   17757,   15282,    5319;
  7:  85, 1409,  9841,  37469,   83823,  109971,   78219,   23247;
  8: 171, 3228, 26280, 120348,  338566,  598260,  647568,  392148,  101655;
  9: 341, 7239, 67428, 361308, 1226054, 2729298, 3981492, 3666924, 1933389, 444447;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(17*n-19+(80*n-90)*x+(142*n-156)*x^2+(112*n-118)*x^3+(33*n-33)*x^4)*P(n-1,x) + (9+65*x+174*x^2+222*x^3+137*x^4+33*x^5)*d/dx 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[(-(17*n - 19 + (80*n - 90)*x + (142*n - 156)*x^2 + (112*n - 118)*x^3 + (33*n - 33)*x^4)*P[n - 1, x] + (9 + 65*x + 174*x^2 + 222*x^3 + 137*x^4 + 33*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*n*x^2)]];
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] = (-(17*n - 19 + (80*n - 90)*x + (142*n - 156)*x^2 + (112*n - 118)*x^3 + (33*n - 33)*x^4)*v[i - 1] + (9 + 65*x + 174*x^2 + 222*x^3 + 137*x^4 + 33*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*n*x^2)); 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)

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