Real-rooted atlas

A121574 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(16\,n + 48\,n x + 48\,n x^{2} + 16\,n x^{3}\right)\,P_{n}(x) = \left(234 + 1860\,x + 5790\,x^{2} + 9240\,x^{3} + 8070\,x^{4} + 3684\,x^{5} + 690\,x^{6} - 299\,n - 2386\,n x - 7653\,n x^{2} - 12668\,n x^{3} - 11461\,n x^{4} - 5394\,n x^{5} - 1035\,n x^{6} + 113\,n^{2} + 782\,n^{2} x + 2343\,n^{2} x^{2} + 3812\,n^{2} x^{3} + 3503\,n^{2} x^{4} + 1710\,n^{2} x^{5} + 345\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(164 + 1852\,x + 8180\,x^{2} + 18860\,x^{3} + 24940\,x^{4} + 19124\,x^{5} + 7932\,x^{6} + 1380\,x^{7} - 78\,n - 838\,n x - 3646\,n x^{2} - 8454\,n x^{3} - 11386\,n x^{4} - 8962\,n x^{5} - 3834\,n x^{6} - 690\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 196\,x + 1184\,x^{2} + 3780\,x^{3} + 7090\,x^{4} + 8108\,x^{5} + 5576\,x^{6} + 2124\,x^{7} + 345\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13710
2104854112
3332395554131240
4109105637305688316113744
5360436820736481125455224192152320
61189173341033953228365563795018461851491688128
73927668534802671886641437589359922554486281141700318709120

All 15 rows

Technical details
Node id
n-36da8aba62cc8b0a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ba5420e9aa19196a

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,      7;
  2:    10,     48,      54;
  3:    33,    239,     555,      413;
  4:   109,   1056,    3730,     5688,      3161;
  5:   360,   4368,   20736,    48112,     54552,     24192;
  6:  1189,  17334,  103395,   322836,    556379,    501846,    185149;
  7:  3927,  66853,  480267,  1886641,   4375893,   5992255,   4486281,   1417003;
  8: 12970, 252516, 2122116, 10051668,  29344536,  54062316,  61382748,  39275100,  10844766;
  9: 42837, 938753, 9034980, 50112324, 176492358, 409279518, 624903732, 605800500, 338393421, 82998377;
%F Formula
The row polynomials satisfy (16*n+48*n*x+48*n*x^2+16*n*x^3)*P(n,x) = (113*n^2-299*n+234+(782*n^2-2386*n+1860)*x+(2343*n^2-7653*n+5790)*x^2+(3812*n^2-12668*n+9240)*x^3+(3503*n^2-11461*n+8070)*x^4+(1710*n^2-5394*n+3684)*x^5+(345*n^2-1035*n+690)*x^6)*P(n-1,x) - (78*n-164+(838*n-1852)*x+(3646*n-8180)*x^2+(8454*n-18860)*x^3+(11386*n-24940)*x^4+(8962*n-19124)*x^5+(3834*n-7932)*x^6+(690*n-1380)*x^7)*d/dx P(n-1,x) + (13+196*x+1184*x^2+3780*x^3+7090*x^4+8108*x^5+5576*x^6+2124*x^7+345*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[((113*n^2 - 299*n + 234 + (782*n^2 - 2386*n + 1860)*x + (2343*n^2 - 7653*n + 5790)*x^2 + (3812*n^2 - 12668*n + 9240)*x^3 + (3503*n^2 - 11461*n + 8070)*x^4 + (1710*n^2 - 5394*n + 3684)*x^5 + (345*n^2 - 1035*n + 690)*x^6)*P[n - 1, x] - (78*n - 164 + (838*n - 1852)*x + (3646*n - 8180)*x^2 + (8454*n - 18860)*x^3 + (11386*n - 24940)*x^4 + (8962*n - 19124)*x^5 + (3834*n - 7932)*x^6 + (690*n - 1380)*x^7)*D[P[n - 1, x], x] + (13 + 196*x + 1184*x^2 + 3780*x^3 + 7090*x^4 + 8108*x^5 + 5576*x^6 + 2124*x^7 + 345*x^8)*D[P[n - 1, x], {x, 2}])/(16*n + 48*n*x + 48*n*x^2 + 16*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] = ((113*n^2 - 299*n + 234 + (782*n^2 - 2386*n + 1860)*x + (2343*n^2 - 7653*n + 5790)*x^2 + (3812*n^2 - 12668*n + 9240)*x^3 + (3503*n^2 - 11461*n + 8070)*x^4 + (1710*n^2 - 5394*n + 3684)*x^5 + (345*n^2 - 1035*n + 690)*x^6)*v[i - 1] - (78*n - 164 + (838*n - 1852)*x + (3646*n - 8180)*x^2 + (8454*n - 18860)*x^3 + (11386*n - 24940)*x^4 + (8962*n - 19124)*x^5 + (3834*n - 7932)*x^6 + (690*n - 1380)*x^7)*deriv(v[i - 1], x) + (13 + 196*x + 1184*x^2 + 3780*x^3 + 7090*x^4 + 8108*x^5 + 5576*x^6 + 2124*x^7 + 345*x^8)*deriv(deriv(v[i - 1], x), x))/(16*n + 48*n*x + 48*n*x^2 + 16*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