Real-rooted atlas

A027467 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(256 + 96\,x + 9\,x^{2} + 48\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
119322
23521149475
3640031685132710108
41146887680019008205281212629
5203161617203205760009504076952434430914
635651584365690881548288034560004276802770272991615663
76207569927486832643839754241083801601814400017962569695721871881835240
8107374182401489816780889841991683071803392650280960870912007185024332424656138436484777
91846835937282899102924802011252654088085779251220734672896351151718439191040027713664112193119683781243899886
10316109592985655405078118404348654387200201125265408060643344384012440803737617557585920167961600010392624037397705904915811697191171
1153876069761024104316165685248914183788953604783519825920016592834396160400246072934468424420556882771476480692841600038106288012341241177147318815445405652
12914793674309632193953851139686418776909823344641097020546744320430516784332800119468407652352240147643760643518970200064372471644160277136640001371826368403895165314416406964239402045
131548112371908608035676953298075648378210009722388482440998277034803210695950330757120335803091779584077654464974028813379654438092817154979725312161404379136010808328960048637480321312659271594323128372265305837338
14261208778387488768650207196201615360749216019259588608529494013611343872256304819088654336898459827783598082350621642457088046592678984417287024318579998728005657205145667789839237124126816512001702311811242408991847829692565231991197680599
Technical details
Node id
n-62389e202e3a3b5d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6ad47d072da00b6d

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:           19,            3;
  2:          352,          114,            9;
  3:         6400,         3168,          513,          27;
  4:       114688,        76800,        19008,        2052,          81;
  5:      2031616,      1720320,       576000,       95040,        7695,        243;
  6:     35651584,     36569088,     15482880,     3456000,      427680,      27702,       729;
  7:    620756992,    748683264,    383975424,   108380160,    18144000,    1796256,     96957,     2187;
  8:  10737418240,  14898167808,   8984199168,  3071803392,   650280960,   87091200,   7185024,   332424,    6561;
  9: 184683593728, 289910292480, 201125265408, 80857792512, 20734672896, 3511517184, 391910400, 27713664, 1121931, 19683;
%F Formula
The row polynomials satisfy (3*n+13+3*x)*P(n,x) = (48*n+256+(9*n+96)*x+9*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[((48*n + 256 + (9*n + 96)*x + 9*x^2)*P[n - 1, x])/(3*n + 13 + 3*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] = ((48*n + 256 + (9*n + 96)*x + 9*x^2)*v[i - 1])/(3*n + 13 + 3*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.