Real-rooted atlas

A038328 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(22 + 8\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 484 + 416\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
048441664964
11064816352576051233272
22342565757443496966963240961233424
35153632191463681789440060006407782403276849006048
41133799046156200968331581444191846409003008082575362621442079892544
524943578881937564211236584970240259477585928215101440123220787284410368209715293936545664
6548758735366012241264641545234006016148627665715265119754649614211769958415766388736838860800167772164497547936000
71207269217792184757307822086352868524032080730598277120470548807680001402087774617622405888081921915622522888153726976134217728227458481036800

All 15 rows

Technical details
Node id
n-2003ecbc129eb9c3 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-ee32e1d26c1bc310

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:             484,               416,                 64;
  1:           10648,             16352,               5760,                512;
  2:          234256,            575744,             349696,              69632,               4096;
  3:         5153632,          19146368,           17894400,            6000640,             778240,              32768;
  4:       113379904,         615620096,          833158144,          419184640,           90030080,            8257536,            262144;
  5:      2494357888,       19375642112,        36584970240,        25947758592,         8215101440,         1232207872,          84410368,          2097152;
  6:     54875873536,      601224126464,      1545234006016,      1486276657152,       651197546496,       142117699584,       15766388736,        838860800,        16777216;
  7:   1207269217792,    18475730782208,     63528685240320,     80730598277120,     47054880768000,     14020877746176,     2240588808192,     191562252288,      8153726976,      134217728;
  8:  26559922791424,   563930077208576,   2561895885389824,   4221837002670080,   3186808347688960,   1245733466406912,   269008238542848,   32866566144000,   2233718538240,    77846282240,   1073741824;
  9: 584318301411328, 17130381698588672, 101863484262481920, 214699669205942272, 205862346796564480, 102729941698740224, 28796444429254656, 4715658067574784, 455032323440640, 25187298836480, 732291923968, 8589934592;
%F Formula
The row polynomials satisfy P(n,x) = (22+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 484 + 416*x + 64*x^2.
%t Mathematica
P[0, x_] := 484 + 416*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 8*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 416*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 8*x)*v[i - 1] + 8*x*deriv(v[i - 1], x)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.