Real-rooted atlas

A038301 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(20 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(279 + 440\,x + 121\,x^{2} + 99\,n + 121\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 279 + 440\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0279440121840
1340292077260133121200
23863714968820255410648014641512000
3419904212503541164203713490146410016105112000000
4442867527713664701261559056124061272585193261201771561275200000
545703926341007975106697606417999046451743303870943597809248018540194871716208000000
6463947993402194548815004350900312979645443959790219030682148112138394345323117947360214358881138240000000
74649045868459308513071990863016564951435797007746246224647840384633625062554438481957291455243858459858023579476913046400000000

All 15 rows

Technical details
Node id
n-9e1ea0bb9ee1e1ae · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-17467381a9b6e863

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:          279,           440,            121;
  1:         3402,          9207,           7260,            1331;
  2:        38637,        149688,         202554,          106480,           14641;
  3:       419904,       2125035,        4116420,         3713490,         1464100,          161051;
  4:      4428675,      27713664,       70126155,        90561240,        61272585,        19326120,         1771561;
  5:     45703926,     341007975,     1066976064,      1799904645,      1743303870,       943597809,       248018540,        19487171;
  6:    463947993,    4021945488,    15004350900,     31297964544,     39597902190,     30682148112,     13839434532,      3117947360,       214358881;
  7:   4649045868,   45930851307,   199086301656,    495143579700,    774624622464,    784038463362,    506255443848,    195729145524,     38584598580,     2357947691;
  8:  46103038191,  511395045480,  2526196821885,   7299831060720,  13616448441750,  17041741694208,  14374038494970,   7955442689040,   2691275750955,   471589538200,   25937424601;
  9: 453281972130, 5578467621111, 30939400251540, 101889938482695, 220819889586780, 329518052290350, 343675124166528, 248465522555910, 120326070671730, 36182707318395, 5706233412220, 285311670611;
%F Formula
The row polynomials satisfy (11*n+20+11*x)*P(n,x) = (99*n+279+(121*n+440)*x+121*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 279 + 440*x + 121*x^2.
%t Mathematica
P[0, x_] := 279 + 440*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((99*n + 279 + (121*n + 440)*x + 121*x^2)*P[n - 1, x])/(11*n + 20 + 11*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 279 + 440*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((99*n + 279 + (121*n + 440)*x + 121*x^2)*v[i - 1])/(11*n + 20 + 11*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. [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.