Real-rooted atlas

A013610 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 20\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141620
216128256400
364768307240968000
42564096245766553665536160000
5102420480163840655360131072010485763200000
6409698304983040524288015728640251658241677721664000000
7163844587525505024367001601468006403523215364697620482684354561280000000
8655362097152293601282348810241174405120375809638475161927688589934592429496729625600000000
9262144943718415099494414092861448455716864338228674569019431321615461882265615461882265668719476736512000000000
10104857641943040754974720805306368056371445760270582939648901943132160206158430208030923764531202748779069440109951162777610240000000000
114194304184549376369098752044291850240354334801920198427489075279370995630082267742732288045354854645760604731395276804837851162214417592186044416204800000000000
1216777216805306368177167400962362232012802126008811520136064563937286349679650406421770330229964854425825574912096757023244288011610842789314568444249301319682814749767106564096000000000000
1367108864348966092883751862272122836064665612283606466560884419665592324716904883159041886761953263616566028585979084812578413021757440201254608348119042195504818343116814636698788954112450359962737049681920000000000000
142684354561503238553639084202393662534723829766878819621273655030556970188833018334182113281509409562610892852829334691381248140878225843683328281756451687366656409827566090715136409827566090715136252201579132747776720575940379279361638400000000000000
Technical details
Node id
n-4826ffe2fa9d46c8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-03d7b8417365c509

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:      4,      16;
  2:     16,     128,       256;
  3:     64,     768,      3072,       4096;
  4:    256,    4096,     24576,      65536,      65536;
  5:   1024,   20480,    163840,     655360,    1310720,     1048576;
  6:   4096,   98304,    983040,    5242880,   15728640,    25165824,    16777216;
  7:  16384,  458752,   5505024,   36700160,  146800640,   352321536,   469762048,    268435456;
  8:  65536, 2097152,  29360128,  234881024, 1174405120,  3758096384,  7516192768,   8589934592,   4294967296;
  9: 262144, 9437184, 150994944, 1409286144, 8455716864, 33822867456, 90194313216, 154618822656, 154618822656, 68719476736;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4+20*x+16*x^2)*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[Together[((4 + 20*x + 16*x^2)*P[n - 1, x])/(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] = ((4 + 20*x + 16*x^2)*v[i - 1])/(1 + x)); 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

Candidate equalities ?

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.