Real-rooted atlas

A086645 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-8 - 52\,x - 100\,x^{2} - 76\,x^{3} - 20\,x^{4} + 4\,n + 26\,n x + 50\,n x^{2} + 38\,n x^{3} + 10\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 16\,x^{2} - 36\,x^{3} - 32\,x^{4} - 10\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2268
384856112
4322887685761664
51281536640010752601624832
651276804352011520014080062976370688
7204836864264192958464184320017694726594565533696
8819217203214909446881280181780482735308821618688690585682608128
932768786432799539244826624151060480312213504385220608258736128723189761233190912
101310723538944412876802724986881118306304295147929649965957125219942400304821043275733401618409324544
11524288157286402070937601572864000761528320024508366848529871667207584350208068668620800354680832007930904576274818138112
1220971526920601610150215688719958016486748979201850568867844881561681928919963402241104850976768882376704000408569249792830535106564102541017088
138388608301989888488216985646808432640296025587712129916049817640514627829769033725509632142755653222401557422250393611124099514368466755570892886974765465661243566063616
1433554432130862284823119003648244712472576172956319744086094297169923101697874329681791545245696157696468713472219331732439040213709452476416138041389744128529525511290889108115685376914256400744448
1513421772856371445761080452710401251446095872977668774297654443542315008222644662173696678583357931520154746101327462426251449461637123258409554018304286749142863052816903212892160005971821849477129538142575001613648205354893312
Technical details
Node id
n-79a292160b221b97 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-800b3c5f1e1db21a

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:      2,       6;
   3:      8,      48,       56;
   4:     32,     288,      768,       576;
   5:    128,    1536,     6400,     10752,       6016;
   6:    512,    7680,    43520,    115200,     140800,      62976;
   7:   2048,   36864,   264192,    958464,    1843200,    1769472,     659456;
   8:   8192,  172032,  1490944,   6881280,   18178048,   27353088,   21618688,    6905856;
   9:  32768,  786432,  7995392,  44826624,  151060480,  312213504,  385220608,  258736128,   72318976;
  10: 131072, 3538944, 41287680, 272498688, 1118306304, 2951479296, 4996595712, 5219942400, 3048210432, 757334016;
%F Formula
The row polynomials satisfy (n-2+(2*n-4)*x+(n-2)*x^2)*P(n,x) = (4*n-8+(26*n-52)*x+(50*n-100)*x^2+(38*n-76)*x^3+(10*n-20)*x^4)*P(n-1,x) - (2*x+16*x^2+36*x^3+32*x^4+10*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (26*n - 52)*x + (50*n - 100)*x^2 + (38*n - 76)*x^3 + (10*n - 20)*x^4)*P[n - 1, x] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*x^5)*D[P[n - 1, x], x])/(n - 2 + (2*n - 4)*x + (n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 8 + (26*n - 52)*x + (50*n - 100)*x^2 + (38*n - 76)*x^3 + (10*n - 20)*x^4)*v[i - 1] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*x^5)*deriv(v[i - 1], x))/(n - 2 + (2*n - 4)*x + (n - 2)*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