Real-rooted atlas

A038221 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115924
222527081576
333756075364572913824
450625121500109350437406561331776
5759375227812527337501640250492075590497962624
611390625410062506150937549207500221433755314410531441191102976
7170859375717609375129169687512916968757750181252790065255580130547829694586471424
825628906251230187500025833937500310007250002325054375011160261000334807830057395628043046721110075314176
9384433593752075941406254982259375006975163125006277646812503766588087501506635235003874204890058113073353874204892641807540224
10576650390625345990234375093417363281251494677812500015694117031250112997642625005649882131250193710244500043584805012558113073350348678440163403380965376
1186497558593755708838867187517126516601562530827729882812536993275859375031074351721875018644611033125079905475856250239716427568754794328551375575319426165313810596091521681143169024
12129746337890625934173632812500308277298828125061655459765625008323487068359375799054758562500055933833099375002876597130825000107872392405937528765971308250051778748354850564859072962028242953648136520347436056576
13194619506835937515180321533203125546491575195312501202281465429687501803422198144531251947695973996093751558156779196875009348940675181250042070233038315625140234110127718753365618643065250550737596137950550737596137952541865828329876488338465357824
14291929260253906252452205786132812509563602565917968752295264615820312500378718661610351562545446239393242187504090161545391796875280468220255437500014724581563410468755889832625364187501766949787609256253855163172965650057827447594484755337918239490902287679245496121035720123168587776
Technical details
Node id
n-c600c5b3860fc422 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b1029f17d1b64336

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:          15,            9;
  2:         225,          270,           81;
  3:        3375,         6075,         3645,          729;
  4:       50625,       121500,       109350,        43740,         6561;
  5:      759375,      2278125,      2733750,      1640250,       492075,        59049;
  6:    11390625,     41006250,     61509375,     49207500,     22143375,      5314410,       531441;
  7:   170859375,    717609375,   1291696875,   1291696875,    775018125,    279006525,     55801305,     4782969;
  8:  2562890625,  12301875000,  25833937500,  31000725000,  23250543750,  11160261000,   3348078300,   573956280,   43046721;
  9: 38443359375, 207594140625, 498225937500, 697516312500, 627764681250, 376658808750, 150663523500, 38742048900, 5811307335, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (15+9*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[(15 + 9*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] = (15 + 9*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.