Real-rooted atlas

A038291 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112214
2144484196
3172886414482744
4207361382434563841638416
5248832207360691201152096032537824
6298598429859841244160276480345602304647529536
73583180841803776209018885806080967680967685376128105413504
8429981696573308928334430208111476736232243203096576258048122882561475789056
951597803527739670528515978035220065812485016453128360755292897286635522764851220661046784
10619173642241031956070407739670528034398535680100329062402006581248278691840265420801658880614401024289254654976
117430083706881362182012928113515167744056757583872018919194624044144787456735746457687588864072990720405504013516820484049565169664
1289161004482561783220089651216346184155136908121341952034054550323209081213419521765791498242522559283226276659201946419209732096294912409656693912375296
1310699320537907223181861165465623181861165465614166692934451259027887226880177083661680643935192481792655865413632819831767047591034880506068992230031366389768192793714773254144
141283918464548864299580975061401632454605631651842163640375443456991668505411584330556168470528826390421176321574076992716822955289477122550587719682125489766412881756165367398413762561638411112006825558016
Technical details
Node id
n-64bc2cdf1204163e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a5035a1c3b326aa3

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:         12,          2;
  2:        144,         48,          4;
  3:       1728,        864,        144,          8;
  4:      20736,      13824,       3456,        384,        16;
  5:     248832,     207360,      69120,      11520,       960,       32;
  6:    2985984,    2985984,    1244160,     276480,     34560,     2304,      64;
  7:   35831808,   41803776,   20901888,    5806080,    967680,    96768,    5376,    128;
  8:  429981696,  573308928,  334430208,  111476736,  23224320,  3096576,  258048,  12288,   256;
  9: 5159780352, 7739670528, 5159780352, 2006581248, 501645312, 83607552, 9289728, 663552, 27648, 512;
%F Formula
The row polynomials satisfy P(n,x) = (12+2*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[(12 + 2*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] = (12 + 2*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.