Real-rooted atlas

A038284 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(38 + 48\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1444 + 3648\,x + 2304\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01444364823047396
154872207936262656110592636056
22085136105354241996185616809984530841654700816
3792351685004326401264250880159694848010085990402548039684704270176
4301093638422819728384720623001601213680844801149802905605809530470412230590464404567235136
5114415582592101167462502438337143685128070977617920101949190963207726675525632325333706342458706834227234792782221696
6434779213849643935583715328194241528004608490715439169536774813851320320782969786597376494507233640448178468776050688281792804290562992179271065856
716521610126284818782462038302729490086082510848279707800326635525299726743030988866943916754075648563738246350110723051816070466764896373139067371521352605460594688257327417311663616

All 15 rows

Technical details
Node id
n-9cc88df18a45b6a2 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f3eacc80e11c3732

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:               1444,                3648,                 2304;
  1:              54872,              207936,               262656,               110592;
  2:            2085136,            10535424,             19961856,             16809984,               5308416;
  3:           79235168,           500432640,           1264250880,           1596948480,            1008599040,             254803968;
  4:         3010936384,         22819728384,          72062300160,         121368084480,          114980290560,           58095304704,           12230590464;
  5:       114415582592,       1011674625024,        3833714368512,        8070977617920,        10194919096320,         7726675525632,         3253337063424,          587068342272;
  6:      4347792138496,      43935583715328,      194241528004608,      490715439169536,       774813851320320,       782969786597376,       494507233640448,       178468776050688,        28179280429056;
  7:    165216101262848,    1878246203830272,     9490086082510848,    27970780032663552,     52997267430309888,     66943916754075648,     56373824635011072,     30518160704667648,      9637313906737152,      1352605460594688;
  8:   6278211847988224,   79303728606167040,   450779088919265280,  1518413773201735680,   3356493603919626240,   5087737673309749248,   5355513340326051840,   3865633689257902080,   1831089642280058880,    513990075025981440,    64925062108545024;
  9: 238572050223552512, 3314895855737782272, 20936184352028098560, 79337119649790689280, 200430618062629109760, 354445724573912530944, 447720915251257933824, 403958720527450767360, 255131823491021537280, 107423925680430120960, 27138675961371820032, 3116402981210161152;
%F Formula
The row polynomials satisfy P(n,x) = (38+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1444 + 3648*x + 2304*x^2.
%t Mathematica
P[0, x_] := 1444 + 3648*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(38 + 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] = 1444 + 3648*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (38 + 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.