Real-rooted atlas

A038326 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 12\,n x\right)\,P_{n}(x) = \left(529 + 552\,x + 144\,x^{2} + 276\,n x + 144\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1232447
252911044322065
3121673808829808691286975
4279841116803213711686359041036803558625
56436343335809205256144036564480119232001492992142559375
614803588992683339218133696801681966080822700800206032896209018885619840625
7340482544724870029352583905036966769913472044151609600165856481283365203968286654464218753609375
87831098528165372648582417906421133442491328157696203097404160010172530851843095987650565274442137638698352648428447890625
91801152661463169151728206965295184535174485950821440512840823253222405264284715827221362314788864545904761241680105589964851597803520322018180859375
104142651121364943227663875112015223655538626402824098418759680322315580401920024215709692805121228333100359680418526983618560921214284595201186749480960068109100646412216423994140625
119528097579139271093659896040333642795387236360880893121124932748801164940597738368001021095758713282566215365487819980826471831713873920776890713341952015012380934144001723160246353928916100448256460677408037109375
122191462443202032127440921027921097611811526877235602882738904783127096320401904506219736960040260347057837998082859068124397191168146124511060584038453605459220594688013811390459412480023779611399684096246084372371865611590930582732817282299169775390625
13504036361936467383683736282279034015232105877602667684192818932530155001799681335216081774459456001504730471286695178241221230527421085941767281871467852437913632056064613915623424103240143684108288002370034602835181568367896136695939072346568824423710721497904875307008645431117038818359375
14115928363245387498091693562176106530406888615077156715828591522397238860999187086336429939578331375944832053835912417146205265924915452872869870915584334966087521212144025617203421342801384570886648665253256573747201907877855282321162243948751867203079372855797580732217425924823253698488565761925877696823296024009361709742431640625
Technical details
Node id
n-971f558d1f43b97e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a59f00ded9d403a5

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:            23,             24;
  2:           529,           1104,            432;
  3:         12167,          38088,          29808,           6912;
  4:        279841,        1168032,        1371168,         635904,         103680;
  5:       6436343,       33580920,       52561440,       36564480,       11923200,        1492992;
  6:     148035889,      926833392,     1813369680,     1681966080,      822700800,      206032896,       20901888;
  7:    3404825447,    24870029352,    58390503696,    67699134720,    44151609600,    16585648128,     3365203968,     286654464;
  8:   78310985281,   653726485824,  1790642113344,  2491328157696,  2030974041600,  1017253085184,   309598765056,   52744421376,   3869835264;
  9: 1801152661463, 16915172820696, 52951845351744, 85950821440512, 84082325322240, 52642847158272, 21362314788864, 5459047612416, 801055899648, 51597803520;
%F Formula
The row polynomials satisfy (23+12*n*x)*P(n,x) = (529+(276*n+552)*x+(144*n+144)*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[((529 + (276*n + 552)*x + (144*n + 144)*x^2)*P[n - 1, x])/(23 + 12*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] = ((529 + (276*n + 552)*x + (144*n + 144)*x^2)*v[i - 1])/(23 + 12*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

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.