Real-rooted atlas

A013620 → basis A008288 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-7 + 6\,x\right)\,P_{n-2}(x) + \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18614
27311436223
3674158410802163554
4622919524207368856129656641
557572225570326448217944673927776902702
653211725018384589892420055220256484898884665614386591
7491816626977524599782326997795246295712174026883452544279936229282754
84545684128496407274452924810577844008938486084605258241415543042379456016796163654137473
942014124829630480948897094432149150432161538099020810124305344425894630411060271361612431361007769658236916814
10388321459330429320970103247312580199595683440243389927520195668405568105107569920372967130888379604224107831347260466176928136531551
113589115719430937200808811705476791602564417973960361329670368034384394246402256019340544102142403827231308378163261957675008713500876836279705614791947588002
12331729069789311936106481213021553577888318897799059605101331948856056154835194624436290873972482411992485388894245050949122540808870912449142755328468008202242176782336235743024876481
1330660525975323123372795340214259410046216038609948078416869156530523936086523897312720077805838630425650935992069772824290155783065683399918243328200580817674243203135207424304749527040130606940163757096450435694
142833842248679731088844208426215409086636856684571460272740248906649538011444812714777633277152130035431011927689854884575209088556392311595187223301698978903047130811810723841547470934691842253187395993619721647964167836416409659877800182094623
Technical details
Node id
n-27a4f67ea9a899c0 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-179c623e61e02598

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,          6;
  2:        73,        114,         36;
  3:       674,       1584,       1080,         216;
  4:      6229,      19524,      20736,        8856,        1296;
  5:     57572,     225570,     326448,      217944,       67392,        7776;
  6:    532117,    2501838,    4589892,     4200552,     2025648,      489888,      46656;
  7:   4918166,   26977524,   59978232,    69977952,    46295712,    17402688,    3452544,     279936;
  8:  45456841,  284964072,  744529248,  1057784400,   893848608,   460525824,  141554304,   23794560,   1679616;
  9: 420141248, 2963048094, 8897094432, 14915043216, 15380990208, 10124305344, 4258946304, 1106027136, 161243136, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = -(7-6*x)*P(n-2,x) + (10+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(7 - 6*x)*P[n - 2, x] + (10 + 6*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; v[2] = 8 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(7 - 6*x)*v[i - 2] + (10 + 6*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

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.