Real-rooted atlas

A200536 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(45 + 57\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1455718120
2202551304869205232414400
3911253462755479654622132191865540458321728000
44100625207765004603635058266540460728812330661673658161329696104976207360000
518452812511686781253329707500562004325062231267254723749657248925069089920692021310128029918160188956824883200000
6830376562563108618750219772996875463731480000660317902875668447736390493286292549267379094556105650864460296788147205626188720646232256340122242985984000000
737366945312533132024843751363644390937534530932559375601029139331257606543524142572185126390685521814371853332887405055627412170469638628384658649172088399187352013963718563213570877376612220032358318080000000

All 15 rows

Technical details
Node id
n-d3d706d2f2257431 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d7967bc76bb8e021

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:               1;
  1:              45,               57,                18;
  2:            2025,             5130,              4869,               2052,                324;
  3:           91125,           346275,            547965,             462213,             219186,              55404,               5832;
  4:         4100625,         20776500,          46036350,           58266540,           46072881,           23306616,            7365816,            1329696,             104976;
  5:       184528125,       1168678125,        3329707500,         5620043250,         6223126725,         4723749657,         2489250690,          899206920,          213101280,           29918160,            1889568;
  6:      8303765625,      63108618750,      219772996875,       463731480000,       660317902875,       668447736390,       493286292549,       267379094556,       105650864460,        29678814720,         5626188720,          646232256,          34012224;
  7:    373669453125,    3313202484375,    13636443909375,     34530932559375,     60102913933125,     76065435241425,     72185126390685,     52181437185333,     28874050556274,     12170469638628,      3846586491720,       883991873520,      139637185632,       13570877376,        612220032;
  8:  16815125390625,  170393270625000,   809218567687500,   2390806912725000,   4918350273243750,   7470367466121000,   8665912947138300,   7831894711954680,   5573006469628641,   3132757884781872,   1386546071542128,    478103517831744,   125909766995040,    24481862786304,    3314559253248,    279172334592,    11019960576;
  9: 756680642578125, 8626159325390625, 46429924228593750, 156778848302062500, 372167690539668750, 659547025979388750, 904307333108508000, 980858914415021700, 853189722763195005, 599609578399201017, 341275889105278002, 156937426306403472, 57875669318944512, 16884403865072352, 3810997151126208, 642166162645248, 76070787856128, 5653239775488, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (45+57*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(45 + 57*x + 18*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (45 + 57*x + 18*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.