Real-rooted atlas

A059473 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(29 + 211\,x + 620\,x^{2} + 950\,x^{3} + 805\,x^{4} + 359\,x^{5} + 66\,x^{6} - 37\,n - 275\,n x - 833\,n x^{2} - 1318\,n x^{3} - 1151\,n x^{4} - 527\,n x^{5} - 99\,n x^{6} + 12\,n^{2} + 84\,n^{2} x + 249\,n^{2} x^{2} + 396\,n^{2} x^{3} + 354\,n^{2} x^{4} + 168\,n^{2} x^{5} + 33\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(27 + 258\,x + 1017\,x^{2} + 2160\,x^{3} + 2685\,x^{4} + 1962\,x^{5} + 783\,x^{6} + 132\,x^{7} - 12\,n - 114\,n x - 452\,n x^{2} - 974\,n x^{3} - 1236\,n x^{4} - 926\,n x^{5} - 380\,n x^{6} - 66\,n x^{7}\right)\,P_{n-1}'(x) + \left(3 + 36\,x + 182\,x^{2} + 508\,x^{3} + 860\,x^{4} + 908\,x^{5} + 586\,x^{6} + 212\,x^{7} + 33\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2187270160
3804809446082112
43562848844010976528427904
51584158406275212307211947245920368640
67048845764195281100864161148012475843990644870144
73136043904026164488602880168528641966822412661120346803264339968

All 15 rows

Technical details
Node id
n-c3af46b2729a4ed8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4584849015d6df29

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,        8;
  2:     18,       72,       70;
  3:     80,      480,      944,       608;
  4:    356,     2848,     8440,     10976,       5284;
  5:   1584,    15840,    62752,    123072,     119472,      45920;
  6:   7048,    84576,   419528,   1100864,    1611480,    1247584,     399064;
  7:  31360,   439040,  2616448,   8602880,   16852864,   19668224,   12661120,    3468032;
  8: 139536,  2232576, 15536064,  61408512,  150781536,  235488000,  228437952,  125839104,   30138640;
  9: 620864, 11175552, 88938240, 410694144, 1212621696, 2373980928, 3081441024, 2557094400, 1230978368, 261917312;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (12*n^2-37*n+29+(84*n^2-275*n+211)*x+(249*n^2-833*n+620)*x^2+(396*n^2-1318*n+950)*x^3+(354*n^2-1151*n+805)*x^4+(168*n^2-527*n+359)*x^5+(33*n^2-99*n+66)*x^6)*P(n-1,x) - (12*n-27+(114*n-258)*x+(452*n-1017)*x^2+(974*n-2160)*x^3+(1236*n-2685)*x^4+(926*n-1962)*x^5+(380*n-783)*x^6+(66*n-132)*x^7)*d/dx P(n-1,x) + (3+36*x+182*x^2+508*x^3+860*x^4+908*x^5+586*x^6+212*x^7+33*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[((12*n^2 - 37*n + 29 + (84*n^2 - 275*n + 211)*x + (249*n^2 - 833*n + 620)*x^2 + (396*n^2 - 1318*n + 950)*x^3 + (354*n^2 - 1151*n + 805)*x^4 + (168*n^2 - 527*n + 359)*x^5 + (33*n^2 - 99*n + 66)*x^6)*P[n - 1, x] - (12*n - 27 + (114*n - 258)*x + (452*n - 1017)*x^2 + (974*n - 2160)*x^3 + (1236*n - 2685)*x^4 + (926*n - 1962)*x^5 + (380*n - 783)*x^6 + (66*n - 132)*x^7)*D[P[n - 1, x], x] + (3 + 36*x + 182*x^2 + 508*x^3 + 860*x^4 + 908*x^5 + 586*x^6 + 212*x^7 + 33*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] = ((12*n^2 - 37*n + 29 + (84*n^2 - 275*n + 211)*x + (249*n^2 - 833*n + 620)*x^2 + (396*n^2 - 1318*n + 950)*x^3 + (354*n^2 - 1151*n + 805)*x^4 + (168*n^2 - 527*n + 359)*x^5 + (33*n^2 - 99*n + 66)*x^6)*v[i - 1] - (12*n - 27 + (114*n - 258)*x + (452*n - 1017)*x^2 + (974*n - 2160)*x^3 + (1236*n - 2685)*x^4 + (926*n - 1962)*x^5 + (380*n - 783)*x^6 + (66*n - 132)*x^7)*deriv(v[i - 1], x) + (3 + 36*x + 182*x^2 + 508*x^3 + 860*x^4 + 908*x^5 + 586*x^6 + 212*x^7 + 33*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