Real-rooted atlas

A027467 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 + n x\right)\,P_{n}(x) = \left(128 + 32\,x + 2\,x^{2} + 16\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116420
225612812396
340963072576327776
4655366553618432204880151632
5104857613107204915208192064001922939328
616777216251658241179648026214403072001843244856687040
7268435456469762048264241152734003201146880010321925017610241088391168
84294967296858993459256371445761879048192367001600440401923211264131072230420815481088
9687194767361546188226561159641169924509715660810569646080158544691215414067294371843317765120396718580736
10109951162777627487790694402319282339840103079215104028185722880050734301184616562688050331648026542080819200112647537653033984
11175921860444164837851162214445354854645760226774273228807086696038400148820616806421703006617622145925120155713536072089600198246424576142818689064960
122814749767106568444249301319688708132091985924837851162214401700807049216004081936918118469449621176328504035246087474249728046137344001903165444718592532482699273223327744
13450359962737049614636698788954112164662861375733761006273041740595239307540692992001061303598710784206364588638208294806555197443109287886848239914188800131952803844907335681107558411468850900580782751744
1472057594037927936252201579132747776307370674568036352204913783045357568880488911523020802641466734569062457782084818698249433809766318081160800811089921074815565824073893570150436641439744124046540825690112245760957856383820873728
Technical details
Node id
n-79531de108dfed13 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-713ec18753f9f2c0

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:          16,            4;
  2:         256,          128,           12;
  3:        4096,         3072,          576,          32;
  4:       65536,        65536,        18432,        2048,          80;
  5:     1048576,      1310720,       491520,       81920,        6400,        192;
  6:    16777216,     25165824,     11796480,     2621440,      307200,      18432,       448;
  7:   268435456,    469762048,    264241152,    73400320,    11468800,    1032192,     50176,    1024;
  8:  4294967296,   8589934592,   5637144576,  1879048192,   367001600,   44040192,   3211264,  131072,   2304;
  9: 68719476736, 154618822656, 115964116992, 45097156608, 10569646080, 1585446912, 154140672, 9437184, 331776, 5120;
%F Formula
The row polynomials satisfy (8+n*x)*P(n,x) = (128+(16*n+32)*x+(2*n+2)*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[((128 + (16*n + 32)*x + (2*n + 2)*x^2)*P[n - 1, x])/(8 + n*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] = ((128 + (16*n + 32)*x + (2*n + 2)*x^2)*v[i - 1])/(8 + n*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

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.