Real-rooted atlas

A038279 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 3\,x + 9\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(81 + 81\,x + 27\,x^{2} + 3\,x^{3} + 81\,n x + 45\,n x^{2} + 6\,n x^{3} + 9\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191221
28121681378
3729291621874326264
46561349923936615552202598496
5590493936605904903499209112587481492992
653144142515287971615629856024603754723923572122021632
74782969446410441004423499920232051667875148803482250423139968318007296
8430467214591650241205308188142851340893002175035712835281015228100776965314414514807808
938742048946490458681394713760419284931008150663523507231849128218741115640814668843046721196830063207309312
10348678440146490458680156905298045247949112960225995285250130173284304492167510101224440064019371024451771470007144929874582769664
113138105960946025554093217259582784953068370272880319621903425021478591910169744916699983030489158406392438068587687765007073479712550916811981009977344
12282429536481451887258369618640349407746368204432745604314895696237533138398947104175408500599646545856582144172595827849531567595400038196790434275499014489813529162718833180672
132541865828329440590076910361982655346096624307991863123525609364405108754846490846013962931827795736841276442033518084038742371678392335216545001489674826926161166923424105081828933124873082193360710270976
142287679245496142703345915927220817881134014514934608861396032706779915043702567850871844195444617628778285523229759566033254484813589805244323268474569340046924757048169676901078380866201552225939373400808107616802529369634876555264
Technical details
Node id
n-ef7c66a14cbd29c0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1e916cb7d7a3299c

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:         9,         12;
  2:        81,        216,          81;
  3:       729,       2916,        2187,         432;
  4:      6561,      34992,       39366,       15552,        2025;
  5:     59049,     393660,      590490,      349920,       91125,       8748;
  6:    531441,    4251528,     7971615,     6298560,     2460375,     472392,      35721;
  7:   4782969,   44641044,   100442349,    99202320,    51667875,   14880348,    2250423,    139968;
  8:  43046721,  459165024,  1205308188,  1428513408,   930021750,  357128352,   81015228,  10077696,   531441;
  9: 387420489, 4649045868, 13947137604, 19284931008, 15066352350, 7231849128, 2187411156, 408146688, 43046721, 1968300;
%F Formula
The row polynomials satisfy (9+(9*n-3)*x+n^2*x^2)*P(n,x) = (81+(81*n+81)*x+(9*n^2+45*n+27)*x^2+(3*n^2+6*n+3)*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[((81 + (81*n + 81)*x + (9*n^2 + 45*n + 27)*x^2 + (3*n^2 + 6*n + 3)*x^3)*P[n - 1, x])/(9 + (9*n - 3)*x + 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] = ((81 + (81*n + 81)*x + (9*n^2 + 45*n + 27)*x^2 + (3*n^2 + 6*n + 3)*x^3)*v[i - 1])/(9 + (9*n - 3)*x + 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

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.