Real-rooted atlas

A094816 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,x^{2} - n x^{2}\right)\,P_{n-2}(x) + \left(2 + n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2224
3410519
48364616106
51611226023265685
6323201160188013383265056
764864448011320145368858195742079
8128224015680565601126721227246675013700390454
9256563251072248192703024118025611362645661441096014000441
1051213824157440989184376252889382721317371211513136534664298641044881660
1110243328046464036595201794643257030624118341200157314800127111404556905869864101547457611
12204878848132352012756480782327043195265928834246081645341280201026328815215866606345488141085051127215589954
134096184320366080042352640317188608161582361657232841601414786560024108210016274461332641965149876478535483601302061345102211815589
14819242598498775041350103041211431936751920998433129306496104587390336235685443136372663111392399425703768272569039496104939213962169267974861548801969976
15163849748482609152041578700844004280323265931468817492331084868510867110419657790472324096685617024607539134820861797239030804042944536432150584567572223697516480525000879886935
Technical details
Node id
n-d6e0d4bc4ae3416f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6f15f544f79dcf9b

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,     2;
   3:   4,    10,      5;
   4:   8,    36,     46,     16;
   5:  16,   112,    260,    232,      65;
   6:  32,   320,   1160,   1880,    1338,     326;
   7:  64,   864,   4480,  11320,   14536,    8858,     1957;
   8: 128,  2240,  15680,  56560,  112672,  122724,    66750,    13700;
   9: 256,  5632,  51072, 248192,  703024, 1180256,  1136264,   566144,  109601;
  10: 512, 13824, 157440, 989184, 3762528, 8938272, 13173712, 11513136, 5346642, 986410;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*x^2*P(n-2,x) + (2+n*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n - 2)*x^2*P[n - 2, x] + (2 + n*x)*P[n - 1, x]];
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 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*x^2*v[i - 2] + (2 + n*x)*v[i - 1]); 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