Real-rooted atlas

A038322 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 88\,x + 264\,n x + 64\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 2904\,x + 2112\,x^{2} + 512\,x^{3} + 2904\,n x + 3520\,n x^{2} + 1024\,n x^{3} + 704\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1113243
21217045761401
313311161619008819240147
4146411703684181763604481024001066033
51610512342560766656099123205632000117964826894139
61771561309217921264982402180710401858560007785676812845056653820457
719487171396829664194807289641978675204770304000299748556898906931213421772815453333859
8214358881498871577628571735808738824683521049466880008792624332843519049728118111600641358954496357219374433
92357947691617353577284040859778561219060727808207794442240021761745223681436128641024584652423168134536495104134217728008110098287947
10259374246017545432611205556182195520191566685798403809564774400047875839492096394935376281602143725551616073995072307201476395008000129922760704181401436840921
11285311670611912997345955274699782850880289744612270080658510482432000965496096423936955743610601472648476979363840298446791639040893218979840001572065404518412369505812484006813143321843
1231384283767211095596815146249860371336316164249587646627840108654229601280001820649781827993621026359433232384171197922552053769848744124088320393016351129600010375631669821441632774767247361161359156838487557757227656081
133452271214393113055862047159361281848273721100860769103346778112172639498144256000325441148501753856429538485564604416408021715415728128281674081948925952140503345528832000494571776261488641167433958581862416607435942789121077521395220481895645983050717787
143797498335832411546617504048108816450386179420793685076744685489356826586482714215424005568659652141121536826861584711863500889764777391460188167229634770022432768432750304228802560019041013386067312645992827654053560321278772567594762241659382948639539298956046499840040709501306383899273
Technical details
Node id
n-82f1a55ac7db116c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-46efc6967b68e78c

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,          32;
  2:        121,         704,          576;
  3:       1331,       11616,        19008,          8192;
  4:      14641,      170368,       418176,        360448,        102400;
  5:     161051,     2342560,      7666560,       9912320,       5632000,       1179648;
  6:    1771561,    30921792,    126498240,     218071040,     185856000,      77856768,      12845056;
  7:   19487171,   396829664,   1948072896,    4197867520,    4770304000,    2997485568,     989069312,    134217728;
  8:  214358881,  4988715776,  28571735808,   73882468352,  104946688000,   87926243328,   43519049728,  11811160064,   1358954496;
  9: 2357947691, 61735357728, 404085977856, 1219060727808, 2077944422400, 2176174522368, 1436128641024, 584652423168, 134536495104, 13421772800;
%F Formula
The row polynomials satisfy (121+(264*n-88)*x+64*n^2*x^2)*P(n,x) = (1331+(2904*n+2904)*x+(704*n^2+3520*n+2112)*x^2+(512*n^2+1024*n+512)*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 + (2904*n + 2904)*x + (704*n^2 + 3520*n + 2112)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*P[n - 1, x])/(121 + (264*n - 88)*x + 64*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 + (2904*n + 2904)*x + (704*n^2 + 3520*n + 2112)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*v[i - 1])/(121 + (264*n - 88)*x + 64*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.