Real-rooted atlas

A038334 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(36 + 48\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1296 + 3456\,x + 2304\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01296345623047056
146656186624248832110592592704
2167961689579521791590415925248530841649787136
360466176403107840107495424014332723209555148802548039684182119424
4217678233617414258688580475289601031956070401031956070405503765708812230590464351298031616
5783641640967313988648962925595459584650132324352086684309913606934744793088308210879692858706834227229509034655744
6282110990745630091839012864140428582060032374476218826752624127031377920665735500136448443823666757632169075682574336281792804290562478758911082496
710155995666841612187194800209926499837226778624202217158166446084044343163328921653924575511052288479329560098242562739026057704243291300868590141441352605460594688208215748530929664

All 15 rows

Technical details
Node id
n-442da0d1b04221aa · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-c91d5d506c107735

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:               1296,                3456,                 2304;
  1:              46656,              186624,               248832,               110592;
  2:            1679616,             8957952,             17915904,             15925248,               5308416;
  3:           60466176,           403107840,           1074954240,           1433272320,             955514880,             254803968;
  4:         2176782336,         17414258688,          58047528960,         103195607040,          103195607040,           55037657088,           12230590464;
  5:        78364164096,        731398864896,        2925595459584,        6501323243520,         8668430991360,         6934744793088,         3082108796928,          587068342272;
  6:      2821109907456,      30091839012864,      140428582060032,      374476218826752,       624127031377920,       665735500136448,       443823666757632,       169075682574336,        28179280429056;
  7:    101559956668416,    1218719480020992,     6499837226778624,    20221715816644608,     40443431633289216,     53924575511052288,     47932956009824256,     27390260577042432,      9130086859014144,     1352605460594688;
  8:   3656158440062976,   48748779200839680,   292492675205038080,  1039973956284579840,   2426605897997352960,   3882569436795764736,   4313966040884183040,   3286831269245091840,   1643415634622545920,   486937965814087680,    64925062108545024;
  9: 131621703842267136, 1930451656353251328, 12869677709021675520, 51478710836086702080, 137276562229564538880, 256249582828520472576, 341666110438027296768, 325396295655264092160, 216930863770176061440, 96413717231189360640, 25710324594983829504, 3116402981210161152;
%F Formula
The row polynomials satisfy P(n,x) = (36+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 3456*x + 2304*x^2.
%t Mathematica
P[0, x_] := 1296 + 3456*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 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] = 1296 + 3456*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 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

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