Real-rooted atlas

A249251 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(48 + 362\,x + 1104\,x^{2} + 1752\,x^{3} + 1532\,x^{4} + 702\,x^{5} + 132\,x^{6} - 77\,n - 562\,n x - 1680\,n x^{2} - 2636\,n x^{3} - 2293\,n x^{4} - 1050\,n x^{5} - 198\,n x^{6} + 29\,n^{2} + 200\,n^{2} x + 576\,n^{2} x^{2} + 884\,n^{2} x^{3} + 761\,n^{2} x^{4} + 348\,n^{2} x^{5} + 66\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(76 + 660\,x + 2420\,x^{2} + 4864\,x^{3} + 5796\,x^{4} + 4100\,x^{5} + 1596\,x^{6} + 264\,x^{7} - 39\,n - 334\,n x - 1213\,n x^{2} - 2424\,n x^{3} - 2881\,n x^{4} - 2038\,n x^{5} - 795\,n x^{6} - 132\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 133\,x + 587\,x^{2} + 1461\,x^{3} + 2245\,x^{4} + 2183\,x^{5} + 1313\,x^{6} + 447\,x^{7} + 66\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 7\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14711
213443794
343216360199818
41429462355259610697108
54693893128892127417504574361772
61549153966360913982017244911314830853536824
7511659233293313805232132355113024927105361657514665224

All 15 rows

Technical details
Node id
n-838b721085bd8a6c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-617f08f2841746a7

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,      7;
  2:    13,     44,      37;
  3:    43,    216,     360,      199;
  4:   142,    946,    2355,     2596,     1069;
  5:   469,   3893,   12889,    21274,    17504,     5743;
  6:  1549,  15396,   63609,   139820,   172449,   113148,    30853;
  7:  5116,  59233,  293313,   805232,  1323551,  1302492,   710536,   165751;
  8: 16897, 223322, 1288999,  4243700,  8715940, 11435366,  9359263,  4368788,   890461;
  9: 55807, 829032, 5464851, 20979732, 51692193, 84769407, 92519733, 64805022, 26433648, 4783807;
%F Formula
The row polynomials satisfy (2*n-2+(6*n-6)*x+(6*n-6)*x^2+(2*n-2)*x^3)*P(n,x) = (29*n^2-77*n+48+(200*n^2-562*n+362)*x+(576*n^2-1680*n+1104)*x^2+(884*n^2-2636*n+1752)*x^3+(761*n^2-2293*n+1532)*x^4+(348*n^2-1050*n+702)*x^5+(66*n^2-198*n+132)*x^6)*P(n-1,x) - (39*n-76+(334*n-660)*x+(1213*n-2420)*x^2+(2424*n-4864)*x^3+(2881*n-5796)*x^4+(2038*n-4100)*x^5+(795*n-1596)*x^6+(132*n-264)*x^7)*d/dx P(n-1,x) + (13+133*x+587*x^2+1461*x^3+2245*x^4+2183*x^5+1313*x^6+447*x^7+66*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 7*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 7*x;
P[n_, x_] := P[n, x] = Expand[Together[((29*n^2 - 77*n + 48 + (200*n^2 - 562*n + 362)*x + (576*n^2 - 1680*n + 1104)*x^2 + (884*n^2 - 2636*n + 1752)*x^3 + (761*n^2 - 2293*n + 1532)*x^4 + (348*n^2 - 1050*n + 702)*x^5 + (66*n^2 - 198*n + 132)*x^6)*P[n - 1, x] - (39*n - 76 + (334*n - 660)*x + (1213*n - 2420)*x^2 + (2424*n - 4864)*x^3 + (2881*n - 5796)*x^4 + (2038*n - 4100)*x^5 + (795*n - 1596)*x^6 + (132*n - 264)*x^7)*D[P[n - 1, x], x] + (13 + 133*x + 587*x^2 + 1461*x^3 + 2245*x^4 + 2183*x^5 + 1313*x^6 + 447*x^7 + 66*x^8)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*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; v[2] = 4 + 7*x; for(n = 2, N, my(i = n + 1); v[i] = ((29*n^2 - 77*n + 48 + (200*n^2 - 562*n + 362)*x + (576*n^2 - 1680*n + 1104)*x^2 + (884*n^2 - 2636*n + 1752)*x^3 + (761*n^2 - 2293*n + 1532)*x^4 + (348*n^2 - 1050*n + 702)*x^5 + (66*n^2 - 198*n + 132)*x^6)*v[i - 1] - (39*n - 76 + (334*n - 660)*x + (1213*n - 2420)*x^2 + (2424*n - 4864)*x^3 + (2881*n - 5796)*x^4 + (2038*n - 4100)*x^5 + (795*n - 1596)*x^6 + (132*n - 264)*x^7)*deriv(v[i - 1], x) + (13 + 133*x + 587*x^2 + 1461*x^3 + 2245*x^4 + 2183*x^5 + 1313*x^6 + 447*x^7 + 66*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*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