Real-rooted atlas

A088996 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 - 2\,x - x^{2} + 2\,n + 8\,n x + 6\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1156
23265584
3151817179351848
410516129182230362150555440
5945175531293864736028615896236452106720
6103952261582037601973813226053549370157102182757596909120
7135135336624935761291210137357737959549154939973918014025858949305755233092480
820270255684508069463306848328410322094584901457925094888998428265089808988012042061737025324451733760
9344594251073798685148215513961189738501486122200079342094856610838476735807970069591977669645913813175785222927206232522711621363200
1065472907522435281090344963598939313497468517618651637309478759232560740282141763435537424134988842746165229551185907913912711465554901315270516771751771506466329600
111374931057551369829276587025288485698825710231973159545773803547828067448026706589432714859440402226049389007601663785948496825665142208776786066705281946926113275258549258359016375152349556104345600
12316234143225127912617844202366307423034022647792901602212199617822395168951068326298619440724162313227567459641189636349737752360247571881622387475936588252988257564363645301081871269130219845611766163090060699734349016262514320858273808486400
13790585358062534413157363882569006957149853908441538969741725470292460941129269942078729789914027151863189556828834264461790720595713248228153494969335403800695282100981319882267343372862075056364973822335649586245628495510184456054647199644925467387954487425221251460727543928465612800
1421345804667687599477383354401502148817049166280052851973266174053228259854417989905349551719553130139051673228523146376000716767333214796788252190141608927299398150010222168920356808950475336675946334816574760720824826634001009105831026220599632839229133861555540305165718000128336186261503879320022245629336375160680029832131217408000
Technical details
Node id
n-fcf7d55270a654e6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-02a5d1db4f001861

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:        1,          5;
  2:        3,         26,          55;
  3:       15,        181,         717,          935;
  4:      105,       1612,        9182,        23036,        21505;
  5:      945,      17553,      129386,       473602,       861589,        623645;
  6:    10395,     226158,     2037601,      9738132,     26053549,      37015710,      21827575;
  7:   135135,    3366249,    35761291,    210137357,    737959549,    1549399739,    1801402585,     894930575;
  8:  2027025,   56845080,   694633068,   4832841032,  20945849014,   57925094888,   99842826508,   98089880120,   42061737025;
  9: 34459425, 1073798685, 14821551396, 118973850148, 612220007934, 2094856610838, 4767358079700, 6959197766964, 5913813175785, 2229272062325;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(8*n-2)*x+(6*n-1)*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[((2*n - 1 + (8*n - 2)*x + (6*n - 1)*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] = ((2*n - 1 + (8*n - 2)*x + (6*n - 1)*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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.