Real-rooted atlas

A038303 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121113
2444121169
3813272613312197
4163522904106481464128561
532880968053240146410161051371293
664211229040212960878460193261217715614826809
7128492881312745360409948013528284248018541948717162748517
82561126421683223851521639792072150848198414832311794736214358881815730721
951225344557568715545659032512324678816119048899228061526243858459858235794769110604499373
101024563201393920204441601967750401298715264595244496018707684160385845985804715895382025937424601137858491849
1120481239043407360562214406184358404761955968261907578241028922628802829537229205187484920205706233412222853116706111792160394037
12409627033681776641499238401855307520163267061761047630312964938828618241697722337520414998793616068474800946646847480094664313842837672123298085122481
13819258572819329024389801984535977728053061795072389119830528214015906790488281561551042697492158504059344827487088890172412306328159913779474634522712143931302875106592253
141638412615684510105699222323215007376384165081140224136191940684885606362716164119806205715215105956087622441541379240961683082758481923211423879291264449666359400300683797498335832413937376385699289
Technical details
Node id
n-f5a3fd4262a598df · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9bb94729b5d21425

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,    11;
  2:   4,    44,    121;
  3:   8,   132,    726,    1331;
  4:  16,   352,   2904,   10648,    14641;
  5:  32,   880,   9680,   53240,   146410,    161051;
  6:  64,  2112,  29040,  212960,   878460,   1932612,    1771561;
  7: 128,  4928,  81312,  745360,  4099480,  13528284,   24801854,   19487171;
  8: 256, 11264, 216832, 2385152, 16397920,  72150848,  198414832,  311794736,  214358881;
  9: 512, 25344, 557568, 7155456, 59032512, 324678816, 1190488992, 2806152624, 3858459858, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (2+11*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 + 11*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 + 11*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.