Real-rooted atlas

A038301 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 22\,x + 11\,n x\right)\,P_{n}(x) = \left(81 + 396\,x + 363\,x^{2} + 99\,n x + 121\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 81 + 396\,x + 363\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
081396363840
172953469801532421200
265616415217641819166473205512000
35904972171026462704312440329422596630612000000
453144177944683572464577623920889440755218052412400927275200000
54782969818419144501305271222576740186782557516436865067812584011558973686208000000
643046721841802544540156632417605105056336208603503944847614428125302436112246104961929229929138240000000
7387420489852325075862503838892237668918256544657937670798831641916759383165772454596725088156267624249235794769103046400000000
8348678440185232507580703168187535305574323472081698690650501437896955448817086121229870136379017526407032043091205212215292190028531167061166560000000000
9313810596098438018250427734850062885378148225296601155452910628502372529976490523383052003514263375380683778402320574220097651050465696340502824585539048934237400473321443840000000000
1028242953648182845997367768353638067915845377787035592015598614293484753660474820871088608949360632566872908222769613446265550394263655378167650682580015252761910864063697639251118564079956889737331129600000000000
112541865828329807748474335668885233217692265309201083164264202781985815301755353444425523966210178153599144330814217103440074620814661387922576952711061403782465465059485771452369834216311896190435764773549560992641483317970015034667648000000000000
12228767924549617828946751253329329494878576873608144851344270242555053021272802057494822195733552681603059191865232101255907861921343174430788914637411600672787473753181297180187380180074964977190850996399983019230073362234253638360898064221894284569624750374861514254080000000000000
1320589113209464975493415101371309688321604675983568416295776230402031357468897438934251011800996424029611824045887877978481515431844516992266606805937862108643665843562718159446579186550505926486202405438173066221128499426898013533013005414137235411061933497786417076899341300606302566835970710650416303104000000000000000
141853020188851841724736784973164489965130793381011607578420455213213760376289626769267211001324539486227820582723462607854428901338166909512271876265708801068815179555859851830129020213718677183760012142235668857730516088830716850078349466244871884681949089404601973097280789374801605536752573643653780096243797823336599047811454076807267376422528000000000000000
Technical details
Node id
n-ba3a4a4a2705dbe3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2a7e84a0ad2b08f0

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:          81,          396,           363;
  1:         729,         5346,          9801,           5324;
  2:        6561,        64152,        176418,         191664,           73205;
  3:       59049,       721710,       2646270,        4312440,         3294225,          966306;
  4:      531441,      7794468,      35724645,       77623920,        88944075,        52180524,        12400927;
  5:     4782969,     81841914,     450130527,     1222576740,      1867825575,      1643686506,       781258401,       155897368;
  6:    43046721,    841802544,    5401566324,    17605105056,     33620860350,     39448476144,     28125302436,     11224610496,      1929229929;
  7:   387420489,   8523250758,   62503838892,   237668918256,    544657937670,    798831641916,    759383165772,    454596725088,    156267624249,     23579476910;
  8:  3486784401,  85232507580,  703168187535,  3055743234720,   8169869065050,  14378969554488,  17086121229870,  13637901752640,   7032043091205,   2122152921900,   285311670611;
  9: 31381059609, 843801825042, 7734850062885, 37814822529660, 115545291062850, 237252997649052, 338305200351426, 337538068377840, 232057422009765, 105046569634050, 28245855390489, 3423740047332;
%F Formula
The row polynomials satisfy (9+(11*n+22)*x)*P(n,x) = (81+(99*n+396)*x+(121*n+363)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 81 + 396*x + 363*x^2.
%t Mathematica
P[0, x_] := 81 + 396*x + 363*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((81 + (99*n + 396)*x + (121*n + 363)*x^2)*P[n - 1, x])/(9 + (11*n + 22)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 81 + 396*x + 363*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((81 + (99*n + 396)*x + (121*n + 363)*x^2)*v[i - 1])/(9 + (11*n + 22)*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

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.