Real-rooted atlas

A103327 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(3 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(4 + 8\,x - 12\,n - 26\,n x + 8\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 4\,x^{2} - 4\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
224832
316019224376
48961920960643840
546081433613440384016036384
622528921601290247168013440384329216
71064965406721013760860160322560430088962887552
849152029818887028736811008047308801290240129024204824764416
92228224157286404472832065601536527155202270822447308803686404608208814592
1099614728021606426738688047710208049201152029520691298402304162201601013760102401737531392
1144040192398458880152410521632086425604055367680314887372814760345603936092165271552027033602252814304573440
12192937984193776844883676364802032140288030482104320291986472961784361779267475865601476034560164003840702873649152116738818048
138388608009261023232445686743041227253350402133747302402438568345601849247662089176717721628677242880524812288049201152017891328106496945752776704
143623878656436207616002315255808007130987888641411341352960187769762611217069978419201056712949760435894091776114708971520178436177921431306240447283202293767614541725696
Technical details
Node id
n-e21ecfeb95d562f5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e63306eb47cd27c0

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       2;
  2:      24,        8;
  3:     160,      192,       24;
  4:     896,     1920,      960,       64;
  5:    4608,    14336,    13440,     3840,      160;
  6:   22528,    92160,   129024,    71680,    13440,      384;
  7:  106496,   540672,  1013760,   860160,   322560,    43008,     896;
  8:  491520,  2981888,  7028736,  8110080,  4730880,  1290240,  129024,   2048;
  9: 2228224, 15728640, 44728320, 65601536, 52715520, 22708224, 4730880, 368640, 4608;
%F Formula
The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (8*n^2-12*n+4+(12*n^2-26*n+8)*x)*P(n-1,x) - ((4*n-2)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 12*n + 4 + (12*n^2 - 26*n + 8)*x)*P[n - 1, x] - ((4*n - 2)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 12*n + 4 + (12*n^2 - 26*n + 8)*x)*v[i - 1] - ((4*n - 2)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023