Real-rooted atlas

A038321 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(18 + 7\,x + 7\,n\right)\,P_{n}(x) = \left(275 + 252\,x + 49\,x^{2} + 77\,n + 49\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 275 + 252\,x + 49\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
027525249576
138725775264634312636
25190910841680850246962401268272
367348618168151897280943250216090168075563728
4853570328286412381531152656192099041251815156117649113374080
510629366041824944769301709462316754532538352097060425148237748235432278819008
613056404575952444960117109845161293631908887243456303644295424905897300118590192576480145304282368
7158625571948225534879118750201624024593067483620374702563610992675493838265101952815307570093389776240353607892616806656
819099376297111103790035802878937207685437504704560043037868096302852458358904128247880761038265101952071339412375726364926028247524917455617552384
92282493364888147065197487674274959163783073892721663915842196556278006627831686830236606548939268141072668837103683016062880610348305875559300993021977326743339194386529280
1027104608708045191729442650592617673829448214119698856585924015517471549422151414890214547040927896436156228439278587271216148126302278955343748165868805126925769350427102576488138412872016555377828081664
113201196944255422466519392432095872368964060193618736106159929158272314898732977102824179821994831321459168253963440120626536700309644996793930210082149772150082054531281083094060842413658637350323886120503496889010407126094620575453184
1237629756236884793137173005370311612085945022917265528497386159299657645903460091826437153373720151663511646128270425915577930042835555548816014776750745787930954409533906732004146776707080413412786860130198144346378212205025244160306225646782230728492415463924417855488
134405098069565595639511244048729029516470158278194135904230080758021042925748056386681616012096397266192835517919340401026541114530691924056388733668539431221666657821001723954253675258605571300106020686042140105038576758255243850261392337602797670175502125183120229669230474756150994346102044212595965952
14513801524838125073493370983791346707222126296667288256520614885909052581073601184422612245892019001675646306166819866881799415635599596334321494464164246578324809686936789442271359049069964740729732800193082876411628963765816873806756076064130764702671641038021008637904570624022381361404017000136729771486358433232930569601877063280142069596160
Technical details
Node id
n-aeb3908f9ea8b1da · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1d0954627bd2ed55

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           275,            252,             49;
  1:          3872,           5775,           2646,            343;
  2:         51909,         108416,          80850,          24696,           2401;
  3:        673486,        1816815,        1897280,         943250,         216090,          16807;
  4:       8535703,       28286412,       38153115,       26561920,        9904125,        1815156,         117649;
  5:     106293660,      418249447,      693017094,      623167545,      325383520,       97060425,       14823774,         823543;
  6:    1305640457,     5952444960,    11710984516,    12936319088,     8724345630,     3644295424,      905897300,      118590192,       5764801;
  7:   15862557194,    82255348791,   187502016240,   245930674836,   203747025636,   109926754938,    38265101952,     8153075700,     933897762,     40353607;
  8:  190993762971,  1110379003580,  2878937207685,  4375047045600,  4303786809630,  2852458358904,  1282478807610,   382651019520,   71339412375,   7263649260,   282475249;
  9: 2282493364888, 14706519748767, 42749591637830, 73892721663915, 84219655627800, 66278316868302, 36606548939268, 14107266883710, 3683016062880, 610348305875, 55930099302, 1977326743;
%F Formula
The row polynomials satisfy (7*n+18+7*x)*P(n,x) = (77*n+275+(49*n+252)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 275 + 252*x + 49*x^2.
%t Mathematica
P[0, x_] := 275 + 252*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((77*n + 275 + (49*n + 252)*x + 49*x^2)*P[n - 1, x])/(7*n + 18 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 275 + 252*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((77*n + 275 + (49*n + 252)*x + 49*x^2)*v[i - 1])/(7*n + 18 + 7*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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.