Real-rooted atlas

A116412 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(83 + 513\,x + 1317\,x^{2} + 1799\,x^{3} + 1380\,x^{4} + 564\,x^{5} + 96\,x^{6} - 104\,n - 667\,n x - 1773\,n x^{2} - 2501\,n x^{3} - 1975\,n x^{4} - 828\,n x^{5} - 144\,n x^{6} + 29\,n^{2} + 188\,n^{2} x + 510\,n^{2} x^{2} + 740\,n^{2} x^{3} + 605\,n^{2} x^{4} + 264\,n^{2} x^{5} + 48\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(93 + 740\,x + 2498\,x^{2} + 4644\,x^{3} + 5141\,x^{4} + 3392\,x^{5} + 1236\,x^{6} + 192\,x^{7} - 39\,n - 316\,n x - 1090\,n x^{2} - 2076\,n x^{3} - 2359\,n x^{4} - 1600\,n x^{5} - 600\,n x^{6} - 96\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 127\,x + 536\,x^{2} + 1278\,x^{3} + 1885\,x^{4} + 1763\,x^{5} + 1022\,x^{6} + 336\,x^{7} + 48\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1459
213342269
34317122598537
4142758150913284364173
546931428383111357364194032433
6154912486417847430874070392408632252069
7511648203194043432628576977460302203404384081959081

All 15 rows

Technical details
Node id
n-cad888bc1baa080e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2e084b6c273aa00d

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:     4,      5;
  2:    13,     34,      22;
  3:    43,    171,     225,       98;
  4:   142,    758,    1509,     1328,      436;
  5:   469,   3142,    8383,    11135,     7364,     1940;
  6:  1549,  12486,   41784,    74308,    74070,    39240,     8632;
  7:  5116,  48203,  194043,   432628,   576977,   460302,   203404,    38408;
  8: 16897, 182206,  857281,  2298688,  3842005,  4098880,  2725912,  1033216,  170896;
  9: 55807, 677757, 3649551, 11436351, 22983714, 30721149, 27311520, 15572520, 5167584, 760400;
%F Formula
The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = (29*n^2-104*n+83+(188*n^2-667*n+513)*x+(510*n^2-1773*n+1317)*x^2+(740*n^2-2501*n+1799)*x^3+(605*n^2-1975*n+1380)*x^4+(264*n^2-828*n+564)*x^5+(48*n^2-144*n+96)*x^6)*P(n-1,x) - (39*n-93+(316*n-740)*x+(1090*n-2498)*x^2+(2076*n-4644)*x^3+(2359*n-5141)*x^4+(1600*n-3392)*x^5+(600*n-1236)*x^6+(96*n-192)*x^7)*d/dx P(n-1,x) + (13+127*x+536*x^2+1278*x^3+1885*x^4+1763*x^5+1022*x^6+336*x^7+48*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[((29*n^2 - 104*n + 83 + (188*n^2 - 667*n + 513)*x + (510*n^2 - 1773*n + 1317)*x^2 + (740*n^2 - 2501*n + 1799)*x^3 + (605*n^2 - 1975*n + 1380)*x^4 + (264*n^2 - 828*n + 564)*x^5 + (48*n^2 - 144*n + 96)*x^6)*P[n - 1, x] - (39*n - 93 + (316*n - 740)*x + (1090*n - 2498)*x^2 + (2076*n - 4644)*x^3 + (2359*n - 5141)*x^4 + (1600*n - 3392)*x^5 + (600*n - 1236)*x^6 + (96*n - 192)*x^7)*D[P[n - 1, x], x] + (13 + 127*x + 536*x^2 + 1278*x^3 + 1885*x^4 + 1763*x^5 + 1022*x^6 + 336*x^7 + 48*x^8)*D[P[n - 1, x], {x, 2}])/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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] = ((29*n^2 - 104*n + 83 + (188*n^2 - 667*n + 513)*x + (510*n^2 - 1773*n + 1317)*x^2 + (740*n^2 - 2501*n + 1799)*x^3 + (605*n^2 - 1975*n + 1380)*x^4 + (264*n^2 - 828*n + 564)*x^5 + (48*n^2 - 144*n + 96)*x^6)*v[i - 1] - (39*n - 93 + (316*n - 740)*x + (1090*n - 2498)*x^2 + (2076*n - 4644)*x^3 + (2359*n - 5141)*x^4 + (1600*n - 3392)*x^5 + (600*n - 1236)*x^6 + (96*n - 192)*x^7)*deriv(v[i - 1], x) + (13 + 127*x + 536*x^2 + 1278*x^3 + 1885*x^4 + 1763*x^5 + 1022*x^6 + 336*x^7 + 48*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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)

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