Real-rooted atlas

A081578 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(81 + 537\,x + 1470\,x^{2} + 2130\,x^{3} + 1725\,x^{4} + 741\,x^{5} + 132\,x^{6} - 115\,n - 768\,n x - 2121\,n x^{2} - 3103\,n x^{3} - 2538\,n x^{4} - 1101\,n x^{5} - 198\,n x^{6} + 36\,n^{2} + 240\,n^{2} x + 666\,n^{2} x^{2} + 984\,n^{2} x^{3} + 816\,n^{2} x^{4} + 360\,n^{2} x^{5} + 66\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(104 + 859\,x + 2999\,x^{2} + 5750\,x^{3} + 6550\,x^{4} + 4439\,x^{5} + 1659\,x^{6} + 264\,x^{7} - 48\,n - 400\,n x - 1411\,n x^{2} - 2736\,n x^{3} - 3154\,n x^{4} - 2164\,n x^{5} - 819\,n x^{6} - 132\,n x^{7}\right)\,P_{n-1}'(x) + \left(16 + 160\,x + 689\,x^{2} + 1671\,x^{3} + 2500\,x^{4} + 2366\,x^{5} + 1385\,x^{6} + 459\,x^{7} + 66\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
27211543
3209013263305
4613668107832792299
51821365404459134266121516985
65474923182733579339015224375319126307
71640172207682418873027568823943611448023247937265

All 15 rows

Technical details
Node id
n-cd9546c47135f724 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7d243563cde7b8a4

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:     7,     21,      15;
  3:    20,     90,     132,      63;
  4:    61,    366,     810,     783,     279;
  5:   182,   1365,    4044,    5913,    4266,     1215;
  6:   547,   4923,   18273,   35793,   39015,    22437,     5319;
  7:  1640,  17220,   76824,  188730,  275688,   239436,   114480,   23247;
  8:  4921,  59052,  307716,  909306, 1666386,  1939140,  1399248,  572427,  101655;
  9: 14762, 199287, 1187880, 4102686, 9047268, 13209318, 12768408, 7879275, 2816694, 444447;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (36*n^2-115*n+81+(240*n^2-768*n+537)*x+(666*n^2-2121*n+1470)*x^2+(984*n^2-3103*n+2130)*x^3+(816*n^2-2538*n+1725)*x^4+(360*n^2-1101*n+741)*x^5+(66*n^2-198*n+132)*x^6)*P(n-1,x) - (48*n-104+(400*n-859)*x+(1411*n-2999)*x^2+(2736*n-5750)*x^3+(3154*n-6550)*x^4+(2164*n-4439)*x^5+(819*n-1659)*x^6+(132*n-264)*x^7)*d/dx P(n-1,x) + (16+160*x+689*x^2+1671*x^3+2500*x^4+2366*x^5+1385*x^6+459*x^7+66*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[((36*n^2 - 115*n + 81 + (240*n^2 - 768*n + 537)*x + (666*n^2 - 2121*n + 1470)*x^2 + (984*n^2 - 3103*n + 2130)*x^3 + (816*n^2 - 2538*n + 1725)*x^4 + (360*n^2 - 1101*n + 741)*x^5 + (66*n^2 - 198*n + 132)*x^6)*P[n - 1, x] - (48*n - 104 + (400*n - 859)*x + (1411*n - 2999)*x^2 + (2736*n - 5750)*x^3 + (3154*n - 6550)*x^4 + (2164*n - 4439)*x^5 + (819*n - 1659)*x^6 + (132*n - 264)*x^7)*D[P[n - 1, x], x] + (16 + 160*x + 689*x^2 + 1671*x^3 + 2500*x^4 + 2366*x^5 + 1385*x^6 + 459*x^7 + 66*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] = ((36*n^2 - 115*n + 81 + (240*n^2 - 768*n + 537)*x + (666*n^2 - 2121*n + 1470)*x^2 + (984*n^2 - 3103*n + 2130)*x^3 + (816*n^2 - 2538*n + 1725)*x^4 + (360*n^2 - 1101*n + 741)*x^5 + (66*n^2 - 198*n + 132)*x^6)*v[i - 1] - (48*n - 104 + (400*n - 859)*x + (1411*n - 2999)*x^2 + (2736*n - 5750)*x^3 + (3154*n - 6550)*x^4 + (2164*n - 4439)*x^5 + (819*n - 1659)*x^6 + (132*n - 264)*x^7)*deriv(v[i - 1], x) + (16 + 160*x + 689*x^2 + 1671*x^3 + 2500*x^4 + 2366*x^5 + 1385*x^6 + 459*x^7 + 66*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