Real-rooted atlas

A038323 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 9\,n x\right)\,P_{n}(x) = \left(121 + 198\,x + 81\,x^{2} + 99\,n x + 81\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1111829
2121396243760
3133165348019291618800
4146419583217641812830432805448000
5161051131769032343303528360180427535429410400000
6177156117393508533664457762392059541075233834043720087236800000
71948717122321668682184325314942604601528220925900261054286446699382637525312000000
8214358881280615262412053701044262989840963362086035026407657584126036547563367210176387420489117760000000
92357947691347261387221704737719084339332375846656930349306535895252044159206069481666769037123835462841138742048902585600000000
102593742460142443058438023440143637356818950876320122043723070501437896955448811437816691070611148646944021095045626054261625379003835462841156320000000000
11285311670611513561007099831513970890215103136632004340210961292736150289975886015508276795163923894184872465700560850833506917352578283354295046409100377313765727153081218560000000000
12313842837672161627320851976415984415750838151267060273032034808613301464755468116707721008608949360632566848806330944947842807750572827255113444467588980030630006249024649707598420656367158397425326214400000000000
13345227121439317343922401527145407797404760894216311896190435765530701891232732597742586150513018124399655100652932116321755418792352803016663828594934055639716306035014600302978701726355409328707690452503650831817935586121596606561152000000000000
143797498335832418699723460270612694000666944314733028366546666100648517280912498408051672484251908778308239469336068756894125590786192134317442061076103826726987124913703262225878056211166468001645118244345540328107240427811140499783548026272587732434315188682441511960320000000000000
Technical details
Node id
n-666262282f291574 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2bd2090585865d9e

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:         11,          18;
  2:        121,         396,          243;
  3:       1331,        6534,         8019,         2916;
  4:      14641,       95832,       176418,       128304,        32805;
  5:     161051,     1317690,      3234330,      3528360,      1804275,       354294;
  6:    1771561,    17393508,     53366445,     77623920,     59541075,     23383404,      3720087;
  7:   19487171,   223216686,    821843253,   1494260460,   1528220925,    900261054,    286446699,     38263752;
  8:  214358881,  2806152624,  12053701044,  26298984096,  33620860350,  26407657584,  12603654756,   3367210176,   387420489;
  9: 2357947691, 34726138722, 170473771908, 433933237584, 665693034930, 653589525204, 415920606948, 166676903712, 38354628411, 3874204890;
%F Formula
The row polynomials satisfy (11+9*n*x)*P(n,x) = (121+(99*n+198)*x+(81*n+81)*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[Together[((121 + (99*n + 198)*x + (81*n + 81)*x^2)*P[n - 1, x])/(11 + 9*n*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] = ((121 + (99*n + 198)*x + (81*n + 81)*x^2)*v[i - 1])/(11 + 9*n*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.