Real-rooted atlas

A013622 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(23 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1231538
25296902251444
3121672380515525337554872
4279841730020714150310500506252085136
56436343209880752737575017853750582187575937579235168
6148035889579270870944463375821272500401709375104793750113906253010936384
734048254471554376834530411720675330562181252155840312584358968751833890625170859375114415582592
87831098528140857905364093262610070012164688270009916865437505174016750001687179375003143812500025628906254347792138496
91801152661463105719830129352757908612070041968174531500410558229112502677553668125011641537687500325384593750053051835937538443359375165216101262848
10414265112136492701728992194507928987259701251378954306035000157380654493125012316746873375006693884170312502494615218750006100961132812588419726562505766503906256278211847988224
119528097579139276835374350252085222892641856046254360942992835687556881865123943750519356159827312503387105390178125015778441258593750514514388867187511185095410156251458925488281258649755859375238572050223552512
122191462443202032117150575642450686061518369152268765013373558511362775001962424346776059375204774714446197500015580684794819375008709699574743750003550149283183593751029028777734375002013317173828125023873326171875001297463378906259065737908494995456
1350403636193646738342733517642439625951672181125138941885039986939948974697250651960977428935281257653454952426631562566551782195014187500434033362141396875002122989271343789062576919901135644531252006606116582031250356906226269531250387941550292968751946195068359375344498040522809827328
14115928363245387498091058476360066581504304487019352456160724751170526787597259319500209931434732117160562527382361052015281812502678709233349321046875199655346585042562500011393375756211667968754953641633135507812501615317923848535156253830793495292968750062458589597167968756266748120117187502919292602539062513090925539866773438464
Technical details
Node id
n-eae515459ec442f4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bbbc90aff2c0cc44

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:            23,             15;
  2:           529,            690,            225;
  3:         12167,          23805,          15525,           3375;
  4:        279841,         730020,         714150,         310500,          50625;
  5:       6436343,       20988075,       27375750,       17853750,        5821875,         759375;
  6:     148035889,      579270870,      944463375,      821272500,      401709375,      104793750,       11390625;
  7:    3404825447,    15543768345,    30411720675,    33056218125,    21558403125,     8435896875,     1833890625,     170859375;
  8:   78310985281,   408579053640,   932626100700,  1216468827000,   991686543750,   517401675000,   168717937500,   31438125000,   2562890625;
  9: 1801152661463, 10571983012935, 27579086120700, 41968174531500, 41055822911250, 26775536681250, 11641537687500, 3253845937500, 530518359375, 38443359375;
%F Formula
The row polynomials satisfy P(n,x) = (23+15*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[(23 + 15*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] = (23 + 15*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.