Real-rooted atlas

A038301 → reverse → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 18\,x + 9\,n x\right)\,P_{n}(x) = \left(121 + 396\,x + 243\,x^{2} + 99\,n x + 81\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 396\,x + 243\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121396243760
1133165348019291618800
2146419583217641812830432805448000
3161051131769032343303528360180427535429410400000
4177156117393508533664457762392059541075233834043720087236800000
51948717122321668682184325314942604601528220925900261054286446699382637525312000000
6214358881280615262412053701044262989840963362086035026407657584126036547563367210176387420489117760000000
72357947691347261387221704737719084339332375846656930349306535895252044159206069481666769037123835462841138742048902585600000000
82593742460142443058438023440143637356818950876320122043723070501437896955448811437816691070611148646944021095045626054261625379003835462841156320000000000
9285311670611513561007099831513970890215103136632004340210961292736150289975886015508276795163923894184872465700560850833506917352578283354295046409100377313765727153081218560000000000
10313842837672161627320851976415984415750838151267060273032034808613301464755468116707721008608949360632566848806330944947842807750572827255113444467588980030630006249024649707598420656367158397425326214400000000000
11345227121439317343922401527145407797404760894216311896190435765530701891232732597742586150513018124399655100652932116321755418792352803016663828594934055639716306035014600302978701726355409328707690452503650831817935586121596606561152000000000000
123797498335832418699723460270612694000666944314733028366546666100648517280912498408051672484251908778308239469336068756894125590786192134317442061076103826726987124913703262225878056211166468001645118244345540328107240427811140499783548026272587732434315188682441511960320000000000000
13417724816941565110253245506747507088084700035239946541640040016658883801277592136874761207527595990156494842082439027116126054305855278099652127702972048582508161629993265343512683971121164501854968493444054288375257925353853442202223529612727488065452121674884879230566200613260284753294258113514384253952000000000000000
1445949729863572161120304747279170748811073505147287307560563742080225535657601873801800749649771004415358425039174733127726877243818555782961032161709749417470080106881517955585985183086368903398453321360054412409141025592456826490789724556411654497835302960009474860265244715932462481604982565396690505800579789427978531584315013432104812975373952000000000000000
Technical details
Node id
n-8c885f3828716061 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2bd2090585865d9e

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:          121,           396,            243;
  1:         1331,          6534,           8019,            2916;
  2:        14641,         95832,         176418,          128304,           32805;
  3:       161051,       1317690,        3234330,         3528360,         1804275,          354294;
  4:      1771561,      17393508,       53366445,        77623920,        59541075,        23383404,         3720087;
  5:     19487171,     223216686,      821843253,      1494260460,      1528220925,       900261054,       286446699,        38263752;
  6:    214358881,    2806152624,    12053701044,     26298984096,     33620860350,     26407657584,     12603654756,      3367210176,      387420489;
  7:   2357947691,   34726138722,   170473771908,    433933237584,    665693034930,    653589525204,    415920606948,    166676903712,    38354628411,     3874204890;
  8:  25937424601,  424430584380,  2344014363735,   6818950876320,  12204372307050,  14378969554488,  11437816691070,   6111486469440,  2109504562605,   426162537900,   38354628411;
  9: 285311670611, 5135610070998, 31513970890215, 103136632004340, 210961292736150, 289975886015508, 276795163923894, 184872465700560, 85083350691735, 25782833542950, 4640910037731, 376572715308;
%F Formula
The row polynomials satisfy (11+(9*n+18)*x)*P(n,x) = (121+(99*n+396)*x+(81*n+243)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 396*x + 243*x^2.
%t Mathematica
P[0, x_] := 121 + 396*x + 243*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((121 + (99*n + 396)*x + (81*n + 243)*x^2)*P[n - 1, x])/(11 + (9*n + 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 396*x + 243*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (99*n + 396)*x + (81*n + 243)*x^2)*v[i - 1])/(11 + (9*n + 18)*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.