Real-rooted atlas

A143685 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(201 + 1329\,x + 3630\,x^{2} + 5250\,x^{3} + 4245\,x^{4} + 1821\,x^{5} + 324\,x^{6} - 289\,n - 1920\,n x - 5277\,n x^{2} - 7687\,n x^{3} - 6264\,n x^{4} - 2709\,n x^{5} - 486\,n x^{6} + 90\,n^{2} + 600\,n^{2} x + 1662\,n^{2} x^{2} + 2448\,n^{2} x^{3} + 2022\,n^{2} x^{4} + 888\,n^{2} x^{5} + 162\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(260 + 2143\,x + 7463\,x^{2} + 14270\,x^{3} + 16210\,x^{4} + 10955\,x^{5} + 4083\,x^{6} + 648\,x^{7} - 120\,n - 1000\,n x - 3523\,n x^{2} - 6816\,n x^{3} - 7834\,n x^{4} - 5356\,n x^{5} - 2019\,n x^{6} - 324\,n x^{7}\right)\,P_{n-1}'(x) + \left(40 + 400\,x + 1721\,x^{2} + 4167\,x^{3} + 6220\,x^{4} + 5870\,x^{5} + 3425\,x^{6} + 1131\,x^{7} + 162\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
213392779
344198288135665
42051230270025658917591
58066045178202578518306510373865
634573111311497522315523984113535131347779239
7141681487646613921613250233085619945449360361858957884905

All 15 rows

Technical details
Node id
n-68623ae8738f73d7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e4990967de006696

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:      2,       3;
  2:     13,      39,       27;
  3:     44,     198,      288,      135;
  4:    205,    1230,     2700,     2565,       891;
  5:    806,    6045,    17820,    25785,     18306,      5103;
  6:   3457,   31113,   114975,   223155,    239841,    135351,     31347;
  7:  14168,  148764,   661392,  1613250,   2330856,   1994544,    936036,    185895;
  8:  59449,  713388,  3706128, 10883430,  19754766,  22692312,  16109442,   6462585,  1121931;
  9: 246410, 3326535, 19776600, 67939830, 148600332, 214571430, 204528240, 124101315, 43499430, 6711903;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (90*n^2-289*n+201+(600*n^2-1920*n+1329)*x+(1662*n^2-5277*n+3630)*x^2+(2448*n^2-7687*n+5250)*x^3+(2022*n^2-6264*n+4245)*x^4+(888*n^2-2709*n+1821)*x^5+(162*n^2-486*n+324)*x^6)*P(n-1,x) - (120*n-260+(1000*n-2143)*x+(3523*n-7463)*x^2+(6816*n-14270)*x^3+(7834*n-16210)*x^4+(5356*n-10955)*x^5+(2019*n-4083)*x^6+(324*n-648)*x^7)*d/dx P(n-1,x) + (40+400*x+1721*x^2+4167*x^3+6220*x^4+5870*x^5+3425*x^6+1131*x^7+162*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[((90*n^2 - 289*n + 201 + (600*n^2 - 1920*n + 1329)*x + (1662*n^2 - 5277*n + 3630)*x^2 + (2448*n^2 - 7687*n + 5250)*x^3 + (2022*n^2 - 6264*n + 4245)*x^4 + (888*n^2 - 2709*n + 1821)*x^5 + (162*n^2 - 486*n + 324)*x^6)*P[n - 1, x] - (120*n - 260 + (1000*n - 2143)*x + (3523*n - 7463)*x^2 + (6816*n - 14270)*x^3 + (7834*n - 16210)*x^4 + (5356*n - 10955)*x^5 + (2019*n - 4083)*x^6 + (324*n - 648)*x^7)*D[P[n - 1, x], x] + (40 + 400*x + 1721*x^2 + 4167*x^3 + 6220*x^4 + 5870*x^5 + 3425*x^6 + 1131*x^7 + 162*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] = ((90*n^2 - 289*n + 201 + (600*n^2 - 1920*n + 1329)*x + (1662*n^2 - 5277*n + 3630)*x^2 + (2448*n^2 - 7687*n + 5250)*x^3 + (2022*n^2 - 6264*n + 4245)*x^4 + (888*n^2 - 2709*n + 1821)*x^5 + (162*n^2 - 486*n + 324)*x^6)*v[i - 1] - (120*n - 260 + (1000*n - 2143)*x + (3523*n - 7463)*x^2 + (6816*n - 14270)*x^3 + (7834*n - 16210)*x^4 + (5356*n - 10955)*x^5 + (2019*n - 4083)*x^6 + (324*n - 648)*x^7)*deriv(v[i - 1], x) + (40 + 400*x + 1721*x^2 + 4167*x^3 + 6220*x^4 + 5870*x^5 + 3425*x^6 + 1131*x^7 + 162*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)

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