Real-rooted atlas

A038320 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(41 + 48\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1681 + 3936\,x + 2304\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01681393623047921
168921242064283392110592704969
22825761132328322323814418137088530841662742241
31158562016781826401587939840185905152010882252802548039685584059449
4475010424133366585888976583001601524422246401338517094406268177612812230590464496981290961
51947542738811596035024976560558642918410937729617920128051468697608994834874368351017946316858706834227244231334895529
67984925229121747856411703043064387247953927175150629355521050022043320320983435279597568575669431959552192558416265216281792804290563936588805702081
73273819343939613449487698980272161536984927856644412717637053644877491626797039616907219045428756487080734013102489635527027800932352103981544783216641352605460594688350356403707485209

All 15 rows

Technical details
Node id
n-42e457d54fd69d5d · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f2fb5cf749ee837b

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1681,                3936,                 2304;
  1:              68921,              242064,               283392,                110592;
  2:            2825761,            13232832,             23238144,              18137088,               5308416;
  3:          115856201,           678182640,           1587939840,            1859051520,            1088225280,             254803968;
  4:         4750104241,         33366585888,          97658300160,          152442224640,          133851709440,           62681776128,           12230590464;
  5:       194754273881,       1596035024976,        5605586429184,        10937729617920,        12805146869760,         8994834874368,         3510179463168,          587068342272;
  6:      7984925229121,      74785641170304,      306438724795392,       717515062935552,      1050022043320320,       983435279597568,       575669431959552,       192558416265216,        28179280429056;
  7:    327381934393961,    3449487698980272,    16153698492785664,     44127176370536448,     77491626797039616,     90721904542875648,     70807340131024896,     35527027800932352,     10398154478321664,      1352605460594688;
  8:  13422659310152401,  157143328509101280,   827877047755265280,   2584591758845706240,   5295261164464373760,   7439196172515803136,   7257752363430051840,   4855360466127421440,   2131621668055941120,    554568238843822080,    64925062108545024;
  9: 550329031716248441, 7087164115760467728, 41485838726402737920, 145706360404926689280, 341166112167633223680, 559179578967437869056, 654649263181390675968, 547441892555866767360, 320453790764409815040, 125055137859281879040, 29281203010953805824, 3116402981210161152;
%F Formula
The row polynomials satisfy P(n,x) = (41+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1681 + 3936*x + 2304*x^2.
%t Mathematica
P[0, x_] := 1681 + 3936*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(41 + 48*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1681 + 3936*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (41 + 48*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.