Real-rooted atlas

A104684 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(4 - 4\,n + n^{2}\right)\,P_{n}(x) = \left(23 + 22\,x - 26\,n - 24\,n x + 7\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(3 + 9\,x + 6\,x^{2} - 2\,n - 6\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
222
316824
411412024258
57681248576642656
65010110408160224016026610
7320168916091200412807680384261720
8201698679224889560566720178080241928962540370
91257472496832079403526549760293888068812871680204824416640
107777314352707846658444867560192396345601341849624514562027524608232904610
1147800080244712520533018880642673920467066880208535040556953608202240552960102402208268120
12292292946166729068041165190005753552640499566144027806929929869798402145792002610432014643202252820835159906
131779856128111935989443089658528049133367360496630886403318806937614770280448429732864077856768079749120378470449152195784325472
1410799942322742394952002265561904324040165500804662372686403641166397441954208256007166411366417485079040268849152023546265695846401064961833469750034
15653364731044873575055441629431189568322088249683241799530572803738771324288235339565260810450263183363229194608646725182464089054085126753976322385510422937617119930193688
Technical details
Node id
n-a16d6c9bf5373530 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3ff6c47fdabd4f1c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:       0;
   2:       2;
   3:      16,        8;
   4:     114,      120,       24;
   5:     768,     1248,      576,       64;
   6:    5010,    11040,     8160,     2240,      160;
   7:   32016,    89160,    91200,    41280,     7680,      384;
   8:  201698,   679224,   889560,   566720,   178080,    24192,     896;
   9: 1257472,  4968320,  7940352,  6549760,  2938880,   688128,   71680,   2048;
  10: 7777314, 35270784, 66584448, 67560192, 39634560, 13418496, 2451456, 202752, 4608;
%F Formula
The row polynomials satisfy (n^2-4*n+4)*P(n,x) = (7*n^2-26*n+23+(6*n^2-24*n+22)*x)*P(n-1,x) - (2*n-3+(6*n-9)*x+(4*n-6)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((7*n^2 - 26*n + 23 + (6*n^2 - 24*n + 22)*x)*P[n - 1, x] - (2*n - 3 + (6*n - 9)*x + (4*n - 6)*x^2)*D[P[n - 1, x], x])/(n^2 - 4*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((7*n^2 - 26*n + 23 + (6*n^2 - 24*n + 22)*x)*v[i - 1] - (2*n - 3 + (6*n - 9)*x + (4*n - 6)*x^2)*deriv(v[i - 1], x))/(n^2 - 4*n + 4)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023