Real-rooted atlas

A038259 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 220\,x + 660\,n x + 400\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 7260\,x + 13200\,x^{2} + 8000\,x^{3} + 7260\,n x + 22000\,n x^{2} + 16000\,n x^{3} + 4400\,n^{2} x^{2} + 8000\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1118091
2121176036005481
3133129040118800128000277171
41464142592026136005632000400000012686161
5161051585640047916000154880000220000000115200000544013451
6177156177304480790614000340736000072600000007603200000313600000022276250041
719487171992074160121754556006559168000018634000000029272320000024147200000081920000000881233896931
821435888112471789440178573348800115441356800040994800000008586547200000106247680000007208960000000207360000000033939028265121
9235794769115433839432025255373616001904782387200081169704000000212517043200000350617344000000356843520000000205286400000000512000000000001279364068775611
102593742460118863581528003472613872200029932294656000014881112400000004675374950400000964197696000000013084262400000000112907520000000005632000000000000123904000000000047387478931259401
112853116706112282493364888046687364281800045272595667200002572306572000000094286728166400000233335842432000000395798937600000000455393664000000000340736000000000000149923840000000000294912000000000001729706521373257491
123138428376721273899203786560616273208519760066399806978560000424430584380000000177797830256640000051333885335040000001044909195264000000015027990912000000000149923840000000000009894973440000000000389283840000000000069222400000000000062358139701786320881
133452271214393132639655117898408011551710756880094951723979340800067437303962600000003178136215837440000010486779432729600000024903669153792000000042980054008320000000053597772800000000000047166040064000000000027833794560000000000098988032000000000000160563200000000000002224283475988175310571
143797498335832413866543760120272010281491362137996001329324135710771200010385344810240400000054381441915440640000020187050408004480000005478807213834240000000110315471954688000000001650811402240000000000018158925424640000000000142880145408000000000007622078464000000000000247267328000000000000036864000000000000000078609533484881054697561
Technical details
Node id
n-ca02ed21329395c7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4e3bd05e67ede8c9

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         11,           80;
  2:        121,         1760,          3600;
  3:       1331,        29040,        118800,         128000;
  4:      14641,       425920,       2613600,        5632000,        4000000;
  5:     161051,      5856400,      47916000,      154880000,      220000000,       115200000;
  6:    1771561,     77304480,     790614000,     3407360000,     7260000000,      7603200000,      3136000000;
  7:   19487171,    992074160,   12175455600,    65591680000,   186340000000,    292723200000,    241472000000,     81920000000;
  8:  214358881,  12471789440,  178573348800,  1154413568000,  4099480000000,   8586547200000,  10624768000000,   7208960000000,   2073600000000;
  9: 2357947691, 154338394320, 2525537361600, 19047823872000, 81169704000000, 212517043200000, 350617344000000, 356843520000000, 205286400000000, 51200000000000;
%F Formula
The row polynomials satisfy (121+(660*n-220)*x+400*n^2*x^2)*P(n,x) = (1331+(7260*n+7260)*x+(4400*n^2+22000*n+13200)*x^2+(8000*n^2+16000*n+8000)*x^3)*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[((1331 + (7260*n + 7260)*x + (4400*n^2 + 22000*n + 13200)*x^2 + (8000*n^2 + 16000*n + 8000)*x^3)*P[n - 1, x])/(121 + (660*n - 220)*x + 400*n^2*x^2)]];
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] = ((1331 + (7260*n + 7260)*x + (4400*n^2 + 22000*n + 13200)*x^2 + (8000*n^2 + 16000*n + 8000)*x^3)*v[i - 1])/(121 + (660*n - 220)*x + 400*n^2*x^2)); 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. [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.