Real-rooted atlas

A133214 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
048113
183618163
216128144321321
3324008004005011683
4641152360032009007218989
5128313614112196009800176498148639
6256819250176100352784002508831361281265729
75122073616588845158450803225401656448518416211462563
8102451200518400184320028224002032128705600115200810020018097453
920481239041548800696960013939200136604166830208174240021780012100242145046719
104096294912446054424780800627264008028979254641664200724483920400387200174242881251595969
11819269222412460032837591042617472004240304643769159681884579845300380881796006543682433633811409933619
121638416056323391897627135180810260490242052098048230861030415076638725771525761282561281603201610599683312439217923848253
13327683686400903168008479744003815884800923444121612825612800106007616005300380800160320160028857628829811600165620044100450144642381823
1465536838860823592960025690112001356759040039074660352656671375366700728320042403046400167518208004104196096610541568529984002508800576005121252055236609
Technical details
Node id
n-a0439c6dbea5d8da · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-806d331b1bf3c1f4

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    4,      8,       1;
  1:    8,     36,      18,       1;
  2:   16,    128,     144,      32,        1;
  3:   32,    400,     800,     400,       50,        1;
  4:   64,   1152,    3600,    3200,      900,       72,       1;
  5:  128,   3136,   14112,   19600,     9800,     1764,      98,       1;
  6:  256,   8192,   50176,  100352,    78400,    25088,    3136,     128,      1;
  7:  512,  20736,  165888,  451584,   508032,   254016,   56448,    5184,    162,     1;
  8: 1024,  51200,  518400, 1843200,  2822400,  2032128,  705600,  115200,   8100,   200,   1;
  9: 2048, 123904, 1548800, 6969600, 13939200, 13660416, 6830208, 1742400, 217800, 12100, 242, 1;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (2*n+4+(3*n+4)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 8*x + x^2.
%t Mathematica
P[0, x_] := 4 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + (3*n + 4)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4 + 8*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 4 + (3*n + 4)*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x))/(n + 2)); 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. [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