Real-rooted atlas

A081580 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(62 + 480\,x + 1470\,x^{2} + 2320\,x^{3} + 2010\,x^{4} + 912\,x^{5} + 170\,x^{6} - 85\,n - 654\,n x - 2029\,n x^{2} - 3268\,n x^{3} - 2895\,n x^{4} - 1342\,n x^{5} - 255\,n x^{6} + 27\,n^{2} + 198\,n^{2} x + 607\,n^{2} x^{2} + 988\,n^{2} x^{3} + 897\,n^{2} x^{4} + 430\,n^{2} x^{5} + 85\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(42 + 486\,x + 2140\,x^{2} + 4880\,x^{3} + 6370\,x^{4} + 4822\,x^{5} + 1976\,x^{6} + 340\,x^{7} - 18\,n - 210\,n x - 942\,n x^{2} - 2198\,n x^{3} - 2942\,n x^{4} - 2286\,n x^{5} - 962\,n x^{6} - 170\,n x^{7}\right)\,P_{n-1}'(x) + \left(3 + 48\,x + 302\,x^{2} + 980\,x^{3} + 1840\,x^{4} + 2088\,x^{5} + 1418\,x^{6} + 532\,x^{7} + 85\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
295461124
3282526765161472
4101121250148268462119216
5342513028940757809123040626242048
611892140215261554726010336359739623592813089344
7408885848740792338556088038801298426410061864317133639237632

All 15 rows

Technical details
Node id
n-bafdd6336a99e435 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c9330823384b8f61

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,       6;
  2:     9,      54,       61;
  3:    28,     252,      676,       516;
  4:   101,    1212,     5014,      8268,      4621;
  5:   342,    5130,    28940,     75780,     91230,      40626;
  6:  1189,   21402,   152615,    547260,   1033635,     973962,     359281;
  7:  4088,   85848,   740792,   3385560,   8803880,   12984264,   10061864,    3171336;
  8: 14121,  338904,  3431532,  19065672,  63322470,  128352168,  154885452,  101841336,   28010041;
  9: 48682, 1314414, 15278088, 100018296, 405144972, 1050315876, 1739818152, 1775314584, 1013754778, 247343646;
%F Formula
The row polynomials satisfy (2*n+6*n*x+6*n*x^2+2*n*x^3)*P(n,x) = (27*n^2-85*n+62+(198*n^2-654*n+480)*x+(607*n^2-2029*n+1470)*x^2+(988*n^2-3268*n+2320)*x^3+(897*n^2-2895*n+2010)*x^4+(430*n^2-1342*n+912)*x^5+(85*n^2-255*n+170)*x^6)*P(n-1,x) - (18*n-42+(210*n-486)*x+(942*n-2140)*x^2+(2198*n-4880)*x^3+(2942*n-6370)*x^4+(2286*n-4822)*x^5+(962*n-1976)*x^6+(170*n-340)*x^7)*d/dx P(n-1,x) + (3+48*x+302*x^2+980*x^3+1840*x^4+2088*x^5+1418*x^6+532*x^7+85*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[((27*n^2 - 85*n + 62 + (198*n^2 - 654*n + 480)*x + (607*n^2 - 2029*n + 1470)*x^2 + (988*n^2 - 3268*n + 2320)*x^3 + (897*n^2 - 2895*n + 2010)*x^4 + (430*n^2 - 1342*n + 912)*x^5 + (85*n^2 - 255*n + 170)*x^6)*P[n - 1, x] - (18*n - 42 + (210*n - 486)*x + (942*n - 2140)*x^2 + (2198*n - 4880)*x^3 + (2942*n - 6370)*x^4 + (2286*n - 4822)*x^5 + (962*n - 1976)*x^6 + (170*n - 340)*x^7)*D[P[n - 1, x], x] + (3 + 48*x + 302*x^2 + 980*x^3 + 1840*x^4 + 2088*x^5 + 1418*x^6 + 532*x^7 + 85*x^8)*D[P[n - 1, x], {x, 2}])/(2*n + 6*n*x + 6*n*x^2 + 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] = ((27*n^2 - 85*n + 62 + (198*n^2 - 654*n + 480)*x + (607*n^2 - 2029*n + 1470)*x^2 + (988*n^2 - 3268*n + 2320)*x^3 + (897*n^2 - 2895*n + 2010)*x^4 + (430*n^2 - 1342*n + 912)*x^5 + (85*n^2 - 255*n + 170)*x^6)*v[i - 1] - (18*n - 42 + (210*n - 486)*x + (942*n - 2140)*x^2 + (2198*n - 4880)*x^3 + (2942*n - 6370)*x^4 + (2286*n - 4822)*x^5 + (962*n - 1976)*x^6 + (170*n - 340)*x^7)*deriv(v[i - 1], x) + (3 + 48*x + 302*x^2 + 980*x^3 + 1840*x^4 + 2088*x^5 + 1418*x^6 + 532*x^7 + 85*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n + 6*n*x + 6*n*x^2 + 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