Real-rooted atlas

A091042 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3 - 6\,x - 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-12 - 62\,x - 112\,x^{2} - 86\,x^{3} - 24\,x^{4} + 8\,n + 40\,n x + 68\,n x^{2} + 48\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 24\,x^{2} - 48\,x^{3} - 40\,x^{4} - 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
241014
3167276164
4644168645681912
5256217667208896424022288
610241075244032876808467231648259808
740965120026112069427210132487683842362243028544
816384237568144998448230409431040108288006753280176320035303296
9655361081344768409630687232752640001159864321095905285800550413160704411525376
10262144484966439321600183271424540753920104676556813285294081065431040489673728982328324797091328
11104857621495808195821568104333312035984506888390344704133871861761442601369610043781120407829913673321984055918994432
124194304943718409542041605720506368225831813126161589862411850848665616061448192015026911641692401721344336005775365472827392651839567872
1316777216411041792456759705630429675520135313489920422955712512952567332864155696444211218324435763201514055761920833440956416274382733312408497397767598398836736
146710886417783848962154194534415788199116878087454720027514487439367103582306304136019328368641931145589555220073313730560148451334553607395451699200222405851545630490660864088573427769344
15268435456765041049610032775168080221936025643682837299201713107265126449884665217024109544566947840182251016945664228557044318208212637269360640142286651392000647260838297601791456362496022758539100161032487537885184
Technical details
Node id
n-47e8524f9cb1ad54 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-dbf3a85c57f4456b

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,      10;
   3:     16,      72,       76;
   4:     64,     416,      864,       568;
   5:    256,    2176,     6720,      8896,      4240;
   6:   1024,   10752,    44032,     87680,     84672,      31648;
   7:   4096,   51200,   261120,    694272,   1013248,     768384,     236224;
   8:  16384,  237568,  1449984,   4823040,   9431040,   10828800,    6753280,    1763200;
   9:  65536, 1081344,  7684096,  30687232,  75264000,  115986432,  109590528,   58005504,  13160704;
  10: 262144, 4849664, 39321600, 183271424, 540753920, 1046765568, 1328529408, 1065431040, 489673728, 98232832;
%F Formula
The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (8*n-12+(40*n-62)*x+(68*n-112)*x^2+(48*n-86)*x^3+(12*n-24)*x^4)*P(n-1,x) - (4*x+24*x^2+48*x^3+40*x^4+12*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (40*n - 62)*x + (68*n - 112)*x^2 + (48*n - 86)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] - (4*x + 24*x^2 + 48*x^3 + 40*x^4 + 12*x^5)*D[P[n - 1, x], x])/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (40*n - 62)*x + (68*n - 112)*x^2 + (48*n - 86)*x^3 + (12*n - 24)*x^4)*v[i - 1] - (4*x + 24*x^2 + 48*x^3 + 40*x^4 + 12*x^5)*deriv(v[i - 1], x))/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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