Real-rooted atlas

A038303 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141115
21688121225
364528145213313375
4256281611616212961464150625
510241408077440212960292820161051759375
6409667584464640170368035138403865224177156111390625
71638431539226019841192576032795840541131364960370819487171170859375
865536144179213877248763248642623667205772067847936593286235894722143588812562890625
9262144648806471368704457949184188904038451948610569523911936112246104967716919716235794769138443359375
1010485762883584035684352026168524801259360256041558888448952391193601496614732801543383943209431790764025937424601576650390625
1141943041268776961744568320143926886407915978752030476518195283810425036816462762060802263629783360207499396808011412466824442853116706118649755859375
12167772165536481288373927936767610060804749587251202089818390528670483400294415804251578368271635574003203319990348928027389920378656136949601893283138428376721129746337890625
136710886423991418883958584115239915723161627442059673601358381953843249807338307584136970180345856282500996963328431598745360640474758619896704356068964922528163198275589492345227121439311946195068359375
142684354561033476505618473392537620320731791361536755341721684521543794688348651368153088109576144276684826366759716577284833905948039168664662067855385666466206785538564569551716505776193327188006013637974983358324129192926025390625
Technical details
Node id
n-c1326eae95374f9d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-432dc7cc45625337

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:      4,      11;
  2:     16,      88,      121;
  3:     64,     528,     1452,      1331;
  4:    256,    2816,    11616,     21296,      14641;
  5:   1024,   14080,    77440,    212960,     292820,     161051;
  6:   4096,   67584,   464640,   1703680,    3513840,    3865224,    1771561;
  7:  16384,  315392,  2601984,  11925760,   32795840,   54113136,   49603708,    19487171;
  8:  65536, 1441792, 13877248,  76324864,  262366720,  577206784,  793659328,   623589472,  214358881;
  9: 262144, 6488064, 71368704, 457949184, 1889040384, 5194861056, 9523911936, 11224610496, 7716919716, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (4+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[(4 + 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] = (4 + 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.