Real-rooted atlas

A038279 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(81 - 36\,x + 108\,n x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(729 + 972\,x + 432\,x^{2} + 64\,x^{3} + 972\,n x + 720\,n x^{2} + 128\,n x^{3} + 144\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191625
281288144513
37293888388810249529
465614665669984368646400166465
5590495248801049760829440288000368642787993
6531441566870414171760149299207776000199065620070445269185
747829695952139217856417623514624016329600062705664126443521048576717709369
843046721612220032214277011233861058562939328000150493593645519667275497472530841611164409217
938742048961987278242479491129645712429056476171136003047495270412290310144305764761642998169626214400170989708825
10348678440161987278240278942752080587731230720714256704000548549148672276531978240917294284801934917632023592960001268776962585050654849
113138105960961367405457630683702728807273173980160101016305280009051060953088547533316915222703033548806385228185601167851520001256089190460397977638653400214585
12282429536481602516344492833138398947104872780877619201363720121280001396449404190729855599704473649038552465408172401161011204204265472000678288162816652298158082835349504572526316648897
132541865828329587453435880483524720615282881021153626814464177283615766400020423072536289281647293093462016956251773075456403418716766208122974765056000264532383498243815944224768331735891968131533373448410608729215641
1422876792454961569377945545696370095664604702411696850634420224223377355865664002859230155080499225944866222026752172125319153582088471793052090368309896407941120083327700801945616026965744025620899361193984165732050534460397977600122664418169862465
Technical details
Node id
n-52d53da180a91748 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-bab60f2685452b19

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,         16;
  2:        81,        288,         144;
  3:       729,       3888,        3888,        1024;
  4:      6561,      46656,       69984,       36864,        6400;
  5:     59049,     524880,     1049760,      829440,      288000,       36864;
  6:    531441,    5668704,    14171760,    14929920,     7776000,     1990656,      200704;
  7:   4782969,   59521392,   178564176,   235146240,   163296000,    62705664,    12644352,    1048576;
  8:  43046721,  612220032,  2142770112,  3386105856,  2939328000,  1504935936,   455196672,   75497472,   5308416;
  9: 387420489, 6198727824, 24794911296, 45712429056, 47617113600, 30474952704, 12290310144, 3057647616, 429981696, 26214400;
%F Formula
The row polynomials satisfy (81+(108*n-36)*x+16*n^2*x^2)*P(n,x) = (729+(972*n+972)*x+(144*n^2+720*n+432)*x^2+(64*n^2+128*n+64)*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[((729 + (972*n + 972)*x + (144*n^2 + 720*n + 432)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(81 + (108*n - 36)*x + 16*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] = ((729 + (972*n + 972)*x + (144*n^2 + 720*n + 432)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(81 + (108*n - 36)*x + 16*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.