Real-rooted atlas

A113413 → 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

\(\left(1 + 3\,x + 3\,x^{2} + x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(57 + 355\,x + 917\,x^{2} + 1259\,x^{3} + 970\,x^{4} + 398\,x^{5} + 68\,x^{6} - 69\,n - 449\,n x - 1209\,n x^{2} - 1725\,n x^{3} - 1376\,n x^{4} - 582\,n x^{5} - 102\,n x^{6} + 18\,n^{2} + 120\,n^{2} x + 334\,n^{2} x^{2} + 496\,n^{2} x^{3} + 414\,n^{2} x^{4} + 184\,n^{2} x^{5} + 34\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(60 + 487\,x + 1672\,x^{2} + 3154\,x^{3} + 3536\,x^{4} + 2359\,x^{5} + 868\,x^{6} + 136\,x^{7} - 24\,n - 200\,n x - 707\,n x^{2} - 1376\,n x^{3} - 1594\,n x^{4} - 1100\,n x^{5} - 419\,n x^{6} - 68\,n x^{7}\right)\,P_{n-1}'(x) + \left(8 + 80\,x + 345\,x^{2} + 839\,x^{3} + 1260\,x^{4} + 1198\,x^{5} + 705\,x^{6} + 235\,x^{7} + 34\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
27201441
3177410650247
4412405225001781481
5997282126308222186348887
62392116776015088164049456225853321
7577597426376643729378881576392228042319927

All 15 rows

Technical details
Node id
n-8be563b1603894fd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0466c0213c6df173

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:    3,     4;
  2:    7,    20,     14;
  3:   17,    74,    106,     50;
  4:   41,   240,    522,    500,     178;
  5:   99,   728,   2126,   3082,    2218,     634;
  6:  239,  2116,   7760,  15088,   16404,    9456,    2258;
  7:  577,  5974,  26376,  64372,   93788,   81576,   39222,    8042;
  8: 1393, 16512,  85260, 250472,  457880,  533360,  386606,  159436,  28642;
  9: 3363, 44908, 265508, 912172, 2006832, 2932044, 2844818, 1767550, 638162, 102010;
%F Formula
The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = (18*n^2-69*n+57+(120*n^2-449*n+355)*x+(334*n^2-1209*n+917)*x^2+(496*n^2-1725*n+1259)*x^3+(414*n^2-1376*n+970)*x^4+(184*n^2-582*n+398)*x^5+(34*n^2-102*n+68)*x^6)*P(n-1,x) - (24*n-60+(200*n-487)*x+(707*n-1672)*x^2+(1376*n-3154)*x^3+(1594*n-3536)*x^4+(1100*n-2359)*x^5+(419*n-868)*x^6+(68*n-136)*x^7)*d/dx P(n-1,x) + (8+80*x+345*x^2+839*x^3+1260*x^4+1198*x^5+705*x^6+235*x^7+34*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[((18*n^2 - 69*n + 57 + (120*n^2 - 449*n + 355)*x + (334*n^2 - 1209*n + 917)*x^2 + (496*n^2 - 1725*n + 1259)*x^3 + (414*n^2 - 1376*n + 970)*x^4 + (184*n^2 - 582*n + 398)*x^5 + (34*n^2 - 102*n + 68)*x^6)*P[n - 1, x] - (24*n - 60 + (200*n - 487)*x + (707*n - 1672)*x^2 + (1376*n - 3154)*x^3 + (1594*n - 3536)*x^4 + (1100*n - 2359)*x^5 + (419*n - 868)*x^6 + (68*n - 136)*x^7)*D[P[n - 1, x], x] + (8 + 80*x + 345*x^2 + 839*x^3 + 1260*x^4 + 1198*x^5 + 705*x^6 + 235*x^7 + 34*x^8)*D[P[n - 1, x], {x, 2}])/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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] = ((18*n^2 - 69*n + 57 + (120*n^2 - 449*n + 355)*x + (334*n^2 - 1209*n + 917)*x^2 + (496*n^2 - 1725*n + 1259)*x^3 + (414*n^2 - 1376*n + 970)*x^4 + (184*n^2 - 582*n + 398)*x^5 + (34*n^2 - 102*n + 68)*x^6)*v[i - 1] - (24*n - 60 + (200*n - 487)*x + (707*n - 1672)*x^2 + (1376*n - 3154)*x^3 + (1594*n - 3536)*x^4 + (1100*n - 2359)*x^5 + (419*n - 868)*x^6 + (68*n - 136)*x^7)*deriv(v[i - 1], x) + (8 + 80*x + 345*x^2 + 839*x^3 + 1260*x^4 + 1198*x^5 + 705*x^6 + 235*x^7 + 34*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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