Real-rooted atlas

A027467 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131619
2996256361
327432230440966859
4811728138244915265536130321
524364806912036864098304010485762476099
67292332831104022118408847360188743681677721647045881
721878164813063681161216061931520198180864352321536268435456893871739
86561279936522547255738368371589120158544691242278584326442450944429496729616983563041
91968394478420155392250822656200658124810701766656380507258888697308774411596411699268719476736322687697779
1059049314928075582720107495424010032906240642105999362853804441608697308774401739461754880206158430208010995116277766131066257801
11177147103926242771366404434186240472979865603531582996481883510931456717527973888019134079303680340161409843203628388371660817592186044416116490258898219
125314413401222499769190417736744960212840939520181624268390411301065588736516620141199361722067137331204081936918118406531099068989446333186975989762814749767106562213314919066161
1315943231105397283537271296691733053449223107379208854183084032629630797086723358030917795841343212367118336397988849516544084904287896862721234971460318003210977524091715584450359962737049642052983462257059
1447829693571283521238044953626411625676838737050992644131952105881633055616847052820148185506775049402486569828352334310633593896968914950229170585617289600444452044823052800592602726418915118434956083272057594037927936799006685782884121
Technical details
Node id
n-11267945ca8c7a9f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e2c46986e009fe93

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:     3,     16;
  2:     9,     96,      256;
  3:    27,    432,     2304,      4096;
  4:    81,   1728,    13824,     49152,      65536;
  5:   243,   6480,    69120,    368640,     983040,     1048576;
  6:   729,  23328,   311040,   2211840,    8847360,    18874368,    16777216;
  7:  2187,  81648,  1306368,  11612160,   61931520,   198180864,   352321536,   268435456;
  8:  6561, 279936,  5225472,  55738368,  371589120,  1585446912,  4227858432,  6442450944,   4294967296;
  9: 19683, 944784, 20155392, 250822656, 2006581248, 10701766656, 38050725888, 86973087744, 115964116992, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (3+16*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[(3 + 16*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] = (3 + 16*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.