Real-rooted atlas

A081577 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 12\,n x + 12\,n x^{2} + 4\,n x^{3}\right)\,P_{n}(x) = \left(64 + 510\,x + 1590\,x^{2} + 2540\,x^{3} + 2220\,x^{4} + 1014\,x^{5} + 190\,x^{6} - 83\,n - 660\,n x - 2111\,n x^{2} - 3488\,n x^{3} - 3153\,n x^{4} - 1484\,n x^{5} - 285\,n x^{6} + 27\,n^{2} + 198\,n^{2} x + 617\,n^{2} x^{2} + 1028\,n^{2} x^{3} + 957\,n^{2} x^{4} + 470\,n^{2} x^{5} + 95\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(42 + 492\,x + 2210\,x^{2} + 5140\,x^{3} + 6830\,x^{4} + 5252\,x^{5} + 2182\,x^{6} + 380\,x^{7} - 18\,n - 210\,n x - 954\,n x^{2} - 2266\,n x^{3} - 3094\,n x^{4} - 2454\,n x^{5} - 1054\,n x^{6} - 190\,n x^{7}\right)\,P_{n-1}'(x) + \left(3 + 48\,x + 304\,x^{2} + 1000\,x^{3} + 1910\,x^{4} + 2208\,x^{5} + 1528\,x^{6} + 584\,x^{7} + 95\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
26364688
316144400336896
4445282248398424769280
512018001035228368368881821695744
6328590442776159072319224327312134056988672
78961881616448077376021105923332736281945698649610207232

All 15 rows

Technical details
Node id
n-9d70b831afe98ad1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-16092db140f6ed9d

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,      6;
  2:    6,     36,      46;
  3:   16,    144,     400,      336;
  4:   44,    528,    2248,     3984,     2476;
  5:  120,   1800,   10352,    28368,    36888,     18216;
  6:  328,   5904,   42776,   159072,   319224,    327312,    134056;
  7:  896,  18816,  164480,   773760,  2110592,   3332736,   2819456,    986496;
  8: 2448,  58752,  601536,  3424896, 11838816,  25405056,  33023424,  23771520,   7259536;
  9: 6688, 180576, 2119296, 14168448, 59378112, 161573184, 285214848, 314817408, 197180704, 53422176;
%F Formula
The row polynomials satisfy (4*n+12*n*x+12*n*x^2+4*n*x^3)*P(n,x) = (27*n^2-83*n+64+(198*n^2-660*n+510)*x+(617*n^2-2111*n+1590)*x^2+(1028*n^2-3488*n+2540)*x^3+(957*n^2-3153*n+2220)*x^4+(470*n^2-1484*n+1014)*x^5+(95*n^2-285*n+190)*x^6)*P(n-1,x) - (18*n-42+(210*n-492)*x+(954*n-2210)*x^2+(2266*n-5140)*x^3+(3094*n-6830)*x^4+(2454*n-5252)*x^5+(1054*n-2182)*x^6+(190*n-380)*x^7)*d/dx P(n-1,x) + (3+48*x+304*x^2+1000*x^3+1910*x^4+2208*x^5+1528*x^6+584*x^7+95*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[((27*n^2 - 83*n + 64 + (198*n^2 - 660*n + 510)*x + (617*n^2 - 2111*n + 1590)*x^2 + (1028*n^2 - 3488*n + 2540)*x^3 + (957*n^2 - 3153*n + 2220)*x^4 + (470*n^2 - 1484*n + 1014)*x^5 + (95*n^2 - 285*n + 190)*x^6)*P[n - 1, x] - (18*n - 42 + (210*n - 492)*x + (954*n - 2210)*x^2 + (2266*n - 5140)*x^3 + (3094*n - 6830)*x^4 + (2454*n - 5252)*x^5 + (1054*n - 2182)*x^6 + (190*n - 380)*x^7)*D[P[n - 1, x], x] + (3 + 48*x + 304*x^2 + 1000*x^3 + 1910*x^4 + 2208*x^5 + 1528*x^6 + 584*x^7 + 95*x^8)*D[P[n - 1, x], {x, 2}])/(4*n + 12*n*x + 12*n*x^2 + 4*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] = ((27*n^2 - 83*n + 64 + (198*n^2 - 660*n + 510)*x + (617*n^2 - 2111*n + 1590)*x^2 + (1028*n^2 - 3488*n + 2540)*x^3 + (957*n^2 - 3153*n + 2220)*x^4 + (470*n^2 - 1484*n + 1014)*x^5 + (95*n^2 - 285*n + 190)*x^6)*v[i - 1] - (18*n - 42 + (210*n - 492)*x + (954*n - 2210)*x^2 + (2266*n - 5140)*x^3 + (3094*n - 6830)*x^4 + (2454*n - 5252)*x^5 + (1054*n - 2182)*x^6 + (190*n - 380)*x^7)*deriv(v[i - 1], x) + (3 + 48*x + 304*x^2 + 1000*x^3 + 1910*x^4 + 2208*x^5 + 1528*x^6 + 584*x^7 + 95*x^8)*deriv(deriv(v[i - 1], x), x))/(4*n + 12*n*x + 12*n*x^2 + 4*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