Real-rooted atlas

A038327 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 13\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121315
2452169225
38156101421973375
4164164056175762856150625
53210401352087880285610371293759375
66424964056035152017136604455516482680911390625
7128582411356812303207997080311886126757532662748517170859375
82561331230284839370243198832016633926454060260810039762728157307212562890625
9512299527787521181107211515795274852668832436156489035786448146831529781060449937338443359375
1010246656019468803374592038385984029941067521621807824060238576320146831529780212089987460137858491849576650390625
1120481464324759040928012801206416640109783914247135954425633131216976010767645517202332989862060303288682067817921603940378649755859375
124096319488114216962474700803619249920376401991682854381770241590298414848646058731032018663918896480363946418481364301184945688823298085122481129746337890625
138192692224269967366434222081045561088012233064729610601989432326891293131008335950540136641213154728271203154202293505125591540429395446057502131845063028751065922531946195068359375
14163841490944629923841637801984292757104643805842360323710696301312275651725240321567769187304326793666478318722207941605453584521877106743574484805029845830848480502984583084393737638569928929192926025390625
Technical details
Node id
n-f5a389a0bc2a9f39 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-dd008bbb0b25b199

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:   2,    13;
  2:   4,    52,    169;
  3:   8,   156,   1014,     2197;
  4:  16,   416,   4056,    17576,     28561;
  5:  32,  1040,  13520,    87880,    285610,    371293;
  6:  64,  2496,  40560,   351520,   1713660,   4455516,    4826809;
  7: 128,  5824, 113568,  1230320,   7997080,  31188612,   67575326,   62748517;
  8: 256, 13312, 302848,  3937024,  31988320, 166339264,  540602608, 1003976272,   815730721;
  9: 512, 29952, 778752, 11811072, 115157952, 748526688, 3243615648, 9035786448, 14683152978, 10604499373;
%F Formula
The row polynomials satisfy P(n,x) = (2+13*x)*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[(2 + 13*x)*P[n - 1, 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] = (2 + 13*x)*v[i - 1]); 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.