Real-rooted atlas

A075504 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-162 + 63\,x + 8\,x^{2}\right)\,P_{n-2}(x) + \left(-81\,x + 72\,x^{2}\right)\,P_{n-2}'(x) + \left(27 - 7\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 108 + 29\,x + x^{2}\)
\(P_{1}(x) = 1296 + 675\,x + 57\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0108291138
112966755712029
21749614985229594134871
32624403346118140558051401684402
442515287669809274031130051012285195114974639
5722759761815756759052211714222061873180230582591359492327
61262703816443487514529840412156412327745525018121534663969033219380296620
722384294920111247075971989994866852816903020532598984601754870674711770639904141264240049483
83998179446482850080350689331822779047112216074834701841150772451311744010548044234794089609801050517987456036451
97168828728456741813381312751124912840458775271362269262110114842151260925725580956443507394331173283947174814201440456051257641182979896
1012878786863533619529653769221053855692001933135227401940170805454613407935186162808010211232382260196890013103972996426167039473063703520463371418824363028006917
1123159221991442005183080953274133113346003769522196598337332990738205291795289053067804146370351976853313736066804160135394585840803476735442185420427012511621112825558321319597630345934303
12416662646579689681383240736916633169465832244976303023142687698273570832301549824776626167405268555038281585055248478438256178131321522898089680424042832327408447853714301056063808882039230380835959112196591720037838834
137498097495038017363705601905848295247516371645953218124483718611956794686518718182080117293947603740171649106400086618941916840000743514847123602204624826213485250973420356012005630044551019076794436701953829495812707604050274010951488932300919737760497
1413494928362011674056995265493461569861865578562290141656278505581509283453221130089344343248933881055829541108718069353197312899814500564218728354403011191051218255876396528921795715640613159143287131843303124161165179344048944344172345839801581910449806517801240120532483142005091074083
Technical details
Node id
n-aba3f426cbb4ac99 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bcfe4e830a38fffc

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           108,             29,               1;
  1:          1296,            675,              57,              1;
  2:         17496,          14985,            2295,             94,              1;
  3:        262440,         334611,           81405,           5805,            140,            1;
  4:       4251528,        7669809,         2740311,         300510,          12285,          195,           1;
  5:      72275976,      181575675,        90522117,       14222061,         873180,        23058,         259,          1;
  6:    1262703816,     4434875145,      2984041215,      641232774,       55250181,      2153466,       39690,        332,        1;
  7:   22384294920,   111247075971,     98999486685,    28169030205,     3259898460,    175487067,     4711770,      63990,      414,      1;
  8:  399817944648,  2850080350689,   3318227790471,  1221607483470,   184115077245,  13117440105,   480442347,    9408960,    98010,    505,   1;
  9: 7168828728456, 74181338131275, 112491284045877, 52713622692621, 10114842151260, 925725580956, 44350739433, 1173283947, 17481420, 144045, 605, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(162-63*x-8*x^2)*P(n-2,x) - (81*x-72*x^2)*d/dx P(n-2,x) + (27-7*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 108 + 29*x + x^2, P(1,x) = 1296 + 675*x + 57*x^2 + x^3.
%t Mathematica
P[0, x_] := 108 + 29*x + x^2;
P[1, x_] := 1296 + 675*x + 57*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(162 - 63*x - 8*x^2)*P[n - 2, x] - (81*x - 72*x^2)*D[P[n - 2, x], x] + (27 - 7*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 108 + 29*x + x^2; v[2] = 1296 + 675*x + 57*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(162 - 63*x - 8*x^2)*v[i - 2] - (81*x - 72*x^2)*deriv(v[i - 2], x) + (27 - 7*x)*v[i - 1] + 9*x*deriv(v[i - 1], x)); 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.