Real-rooted atlas

A038338 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(60 + 108\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3600 + 12960\,x + 11664\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03600129601166428224
12160001166400209952012597124741632
21296000093312000251942400302330880136048896796594176
3777600000699840000025194240000453496320004081466880014693280768133827821568
4466560000005038848000002267481600000544195584000073466403840005289581076480158687432294422483074023424
5279936000000035271936000000190468454400000571405363200000102852965376000011108120260608006664872156364801713824268779523777156435935232
6167961600000000241864704000000015237476352000000548549148672000001234235584512000001777299241697280001599569317527552008226356490141696018509302102818816634562281237118976
710077696000000000163258675200000000117546246144000000049369423380480000001332974431272960000023993539762913280000287922477154959360002221116252338257920099950231355221606401999004627104432128106606463247835987968

All 15 rows

Technical details
Node id
n-bd16f85c006e2508 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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:                 3600,                 12960,                  11664;
  1:               216000,               1166400,                2099520,                 1259712;
  2:             12960000,              93312000,              251942400,               302330880,                136048896;
  3:            777600000,            6998400000,            25194240000,             45349632000,              40814668800,              14693280768;
  4:          46656000000,          503884800000,          2267481600000,           5441955840000,            7346640384000,            5289581076480,            1586874322944;
  5:        2799360000000,        35271936000000,        190468454400000,         571405363200000,         1028529653760000,         1110812026060800,          666487215636480,          171382426877952;
  6:      167961600000000,      2418647040000000,      15237476352000000,       54854914867200000,       123423558451200000,       177729924169728000,       159956931752755200,        82263564901416960,        18509302102818816;
  7:    10077696000000000,    163258675200000000,    1175462461440000000,     4936942338048000000,     13329744312729600000,     23993539762913280000,     28792247715495936000,     22211162523382579200,      9995023135522160640,      1999004627104432128;
  8:   604661760000000000,  10883911680000000000,   88159684608000000000,   423166486118400000000,   1332974431272960000000,   2879224771549593600000,   4318837157324390400000,   4442232504676515840000,   2998506940656648192000,   1199402776262659276800,    215892499727278669824;
  9: 36279705600000000000, 718338170880000000000, 6465043537920000000000, 34911235104768000000000, 125680446377164800000000, 316714724870455296000000, 570086504766819532800000, 732968363271625113600000, 659671526944462602240000, 395802916166677561344000, 142489049820003922083840, 23316389970546096340992;
%F Formula
The row polynomials satisfy P(n,x) = (60+108*x)*P(n-1,x) for n >= 1, with P(0,x) = 3600 + 12960*x + 11664*x^2.
%t Mathematica
P[0, x_] := 3600 + 12960*x + 11664*x^2;
P[n_, x_] := P[n, x] = Expand[(60 + 108*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] = 3600 + 12960*x + 11664*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 108*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.