Real-rooted atlas

A038305 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 + 195\,x + 36\,x^{2} + 117\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 4563\,x + 1989\,x^{2} + 243\,x^{3} + 1521\,n x + 1053\,n x^{2} + 162\,n x^{3} + 117\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 169 + 312\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
016931281562
121976084315943211872
22856110545682134224642025240640
33712931713660177957073008013162587484734976
4482680926733096347016151898208051333756823443572191095040
5627485174054519566315693934318423201557123753104665232506111399681721761792
68157307216023857632109472028128982320256404852175010762839361690317721455667253144132078036992
7106044993738809891786818297467557217515524499294735408950314813051286592239108851565312621785971968300590558003200
81378584918491272539924760297333847804532528831212802052600527250818513933328214247771010369011635204041608805255879000714492910763188043776
9179216039403718197320924068472430447067155814528579288041931696485250195079154109846127486250886131921659584019265001970518295348500102172484725509168194476119162880
10232980851224812580710967413287369914974247541007851620409920817668081462375434747829159072159314642523036411595577902087513350768495951358122000796945380663979430208898135293487650883305472
1130287510659225336345012791070361132286936952576617032692384927648153539895296823759184047890985396384631065519901211593275444241922539512559751314019488065450044894589777183362618525761517848640131248730862135601108877312
123937376385699289508830179074985041717301854378074512818136376415301762794426094402192251857218573510380088750356740577752330142516155028992770318809791230714630936558238002042703834861692039988572294413812422624915687269005610761680251100567158938664960
135118589301409075770872774942587202025759527815671117654579471611674865360495375534916752262536215762183452411561895910627125179665734723831278831680214588811298985695475505438142735007966544955960591994494428993520897807470619155545087280460209852764875367332019219392499995457355776
146654166091831798419827691458705425344382712984689970890807327154578679784576085865092718903725500684807139468918327683943494104420373703216980284100666332160557930909377362807014129304447669840002761735584733004884137096824613043246685988472195803844593847785602182468754700076405059993612440502369340111843859019857920
Technical details
Node id
n-541403f60ed1ee8c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cbbf8868a78f08ff

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           169,            312,             81;
  1:          2197,           6084,           3159,            432;
  2:         28561,         105456,          82134,          22464,           2025;
  3:        371293,        1713660,        1779570,         730080,         131625,           8748;
  4:       4826809,       26733096,       34701615,       18982080,        5133375,         682344,         35721;
  5:      62748517,      405451956,      631569393,      431842320,      155712375,       31046652,       3250611,        139968;
  6:     815730721,     6023857632,    10947202812,     8982320256,     4048521750,     1076283936,     169031772,      14556672,       531441;
  7:   10604499373,    88098917868,   182974675572,   175155244992,    94735408950,    31481305128,    6592239108,     851565312,     62178597,     1968300;
  8:  137858491849,  1272539924760,  2973338478045,  3252883121280,  2052600527250,   818513933328,  214247771010,   36901163520,   4041608805,   255879000,    7144929;
  9: 1792160394037, 18197320924068, 47243044706715, 58145285792880, 41931696485250, 19507915410984, 6127486250886, 1319216595840, 192650019705, 18295348500, 1021724847, 25509168;
%F Formula
The row polynomials satisfy (169+(117*n+195)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (2197+(1521*n+4563)*x+(117*n^2+1053*n+1989)*x^2+(27*n^2+162*n+243)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 312*x + 81*x^2.
%t Mathematica
P[0, x_] := 169 + 312*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2197 + (1521*n + 4563)*x + (117*n^2 + 1053*n + 1989)*x^2 + (27*n^2 + 162*n + 243)*x^3)*P[n - 1, x])/(169 + (117*n + 195)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 169 + 312*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (1521*n + 4563)*x + (117*n^2 + 1053*n + 1989)*x^2 + (27*n^2 + 162*n + 243)*x^3)*v[i - 1])/(169 + (117*n + 195)*x + (9*n^2 + 36*n + 36)*x^2)); 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.