Real-rooted atlas

A038277 → x ↦ x+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(29 + 33\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1293362
2841191410893844
324389832599474335937238328
4707281321934854950944168692118592114776336
52051114911670136526559621030223017017195854539135393916132832
659482332140612075021155343513517529349860149603934156809558382129146796956800235584
7172498763091374041871514690694664818896145053951012319954415691170175773262167997707426184429773521614606208
85002464129614553967345576181373527039324127811305032858714557356070534504935931123041148773401298874787706641406408618241218340105584896
914507145975869148573184649417676264150818036179559791768926830648998939868543487644706950558264579943285904412903159795716523670726493609014641148440195313537086546263552
104207072333002014787358172036770245145754671538057438905658998396014813682820936461020228339300313236419182045888228069012473054469192636053225534157330645134593304765663701531578985264449839299365868340224
11122005097657058291527167256879729638689055082246737552966263631525610405675080688554104437010754733727999870686122381452766895080229947260939181127210565964846539615858521467632110123360154885736962993592315054210651372681752036560683837093888
12353814783205469041483140186721950828430237911686218646674114695527085656935660293660099521035430095534663905334850714104709812426047991465276692329361367006453816492389416489465796895249024532477430977740850118231560885061941758865306677693231616678895149529849613226266762397899821056
13102606287129586021891517865419951462185891036335700518584526918432402137112926647438212301095279936706349535251960365389048399021513822846923144182605843643501361539226905514772371261620033057210858832347056218599786965199510683152443281788945046331546110308745174156662879434713727533029755040353993448503713200028539268669788905472
1429755823267579946348147404104653868742113183506269120087877649405915959569787986201714537249942446836543027779112111366212038661516666970341940094813495672672465627252307896927516051985677625122036288903537934780831905809649503027015741994108433997299310157792216944869240261783513569193212764525246886689205029122346383721340092507478181633168178380062252912401769434657526912139264
Technical details
Node id
n-37e8f83dfbff7f8a · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f350b4cb45bc0810

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:             29,              33;
  2:            841,            1914,            1089;
  3:          24389,           83259,           94743,            35937;
  4:         707281,         3219348,         5495094,          4168692,          1185921;
  5:       20511149,       116701365,       265596210,        302230170,        171958545,         39135393;
  6:      594823321,      4061207502,     11553435135,      17529349860,      14960393415,       6809558382,       1291467969;
  7:    17249876309,    137404187151,    469069466481,     889614505395,    1012319954415,     691170175773,     262167997707,      42618442977;
  8:   500246412961,   4553967345576,  18137352703932,   41278113050328,   58714557356070,   53450493593112,   30411487734012,    9887478770664,   1406408618241;
  9: 14507145975869, 148573184649417, 676264150818036, 1795597917689268, 3064899893986854, 3487644706950558, 2645799432859044, 1290315979571652, 367072649360901, 46411484401953;
%F Formula
The row polynomials satisfy P(n,x) = (29+33*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[(29 + 33*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] = (29 + 33*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.