Real-rooted atlas

A172431 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n}(x) = \left(-9 - 58\,x - 128\,x^{2} - 118\,x^{3} - 39\,x^{4} + 21\,n + 110\,n x + 196\,n x^{2} + 146\,n x^{3} + 39\,n x^{4}\right)\,P_{n-1}(x) + \left(-5 - 43\,x - 138\,x^{2} - 206\,x^{3} - 145\,x^{4} - 39\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13710
28364084
321139303217680
4554801554220811595456
5144156066961421614904616043680
6377487826079736681158319598232689349504
798714849950313351257027058752355988451733832796160
82584443163302961396548366144060906606272120365209291948022369536
967651302651108092546276417191422357944704927638043217548218931814875913178956800
101771137833036166832036776874800222187095132322621390378486552288921267129494298258560711431655424
114636810881121154779273136384307034208896779072185876409627327833282791483776188573883275773419213710928011453245440
1212139331043163621409825477086012035003354020370296973535161217211627576220405015351992820206012070261138439488207672706032191625967616
133178118796379111878730865595122454411438517088401457473459005089782596446015068583702917083842548513851267666676021310274253030920873855438727733007749120
14832040247811403412783122879942184166334566326953969502021696743532051309212702492391083080812602402762041281403021080941482400040472329745160144771660180204445282005864062009344
Technical details
Node id
n-7b6bb841dd0d667f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3996b27c790d41f9

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:    8,     36,      40;
  3:   21,    139,     303,     217;
  4:   55,    480,    1554,    2208,     1159;
  5:  144,   1560,    6696,   14216,    14904,     6160;
  6:  377,   4878,   26079,   73668,   115831,    95982,    32689;
  7:  987,  14849,   95031,  335125,   702705,   875235,   598845,   173383;
  8: 2584,  44316,  330296, 1396548,  3661440,  6090660,  6272120,  3652092,   919480;
  9: 6765, 130265, 1108092, 5462764, 17191422, 35794470, 49276380, 43217548, 21893181, 4875913;
%F Formula
The row polynomials satisfy (4*n+8*n*x+4*n*x^2)*P(n,x) = (21*n-9+(110*n-58)*x+(196*n-128)*x^2+(146*n-118)*x^3+(39*n-39)*x^4)*P(n-1,x) - (5+43*x+138*x^2+206*x^3+145*x^4+39*x^5)*d/dx 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[((21*n - 9 + (110*n - 58)*x + (196*n - 128)*x^2 + (146*n - 118)*x^3 + (39*n - 39)*x^4)*P[n - 1, x] - (5 + 43*x + 138*x^2 + 206*x^3 + 145*x^4 + 39*x^5)*D[P[n - 1, x], x])/(4*n + 8*n*x + 4*n*x^2)]];
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] = ((21*n - 9 + (110*n - 58)*x + (196*n - 128)*x^2 + (146*n - 118)*x^3 + (39*n - 39)*x^4)*v[i - 1] - (5 + 43*x + 138*x^2 + 206*x^3 + 145*x^4 + 39*x^5)*deriv(v[i - 1], x))/(4*n + 8*n*x + 4*n*x^2)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023