Real-rooted atlas

A038333 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(361 + 665\,x + 196\,x^{2} + 399\,n x + 196\,n x^{2} + 49\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6859 + 22743\,x + 15827\,x^{2} + 3087\,x^{3} + 7581\,n x + 8379\,n x^{2} + 2058\,n x^{3} + 931\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 361 + 1064\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
036110644411866
168593032425137548867808
2130321768208955206417088600252330848
32476099182449403024819019811680570237560505277088336
4470458814159846328620734157528438403250353756897592857648012477723872
5893871739922099267622931152839250320576801440990162545868992127667185335270675277894301056
6169835630412002272695365809225385887609745534725475762617502324028934085827060850880114263044669488812405836063488
732268769777942798578863321419110772836421687774773952187271081518509935223693192332142468495668497694899279848258651403536070073234045184768
8613106625780190352555378120337038808548645588668172435840593025091475250377538500341296157767672535410433818734361607585584571845766718533000341795051292202290223018496
9116490258898219188836840740270878267901096296451537895600488632017706034874046750131509244285551446594688711980138226670288703936052846239183853580122086698500714351657196128473505099265544968412598272
102213314919066161391407269898015841784508144995559063896002187904534405046219939103323754283443956729389762505981710552452441033616516489948163012235633479649560892785890860008143608892035546491959162617623391775369691933424542068637696
11420529834622570598056466305400826044007031925580937162962312540412419996813849070277316899625132251332164019908848842535464377939324425505465955028659214880444029389468533760129528760605006704904654442594680175695658327365777768516313431899024603977256589371300377530368
127990066857828841211648476951720476712888822541017044107091232704668863367017408368385269376629530025390876159506992175016294014304190566482523161692077062910890496659699685302933104832000388909300641860044587615952043254097108911681705025927684432126696861950514054465793521526001913910251645004354934077349888
13151811270298747982993355828080288113308201947263399219813116995552673588550496666344095445274338490378233751113997054594927698795693104529660346052798955760280245366200473920268592014730479906966595018473191780488350025414941092664654831350650995732148309680730021052036201880571982527613755766043491054546442125121537574654540847528964439090777849856
142884414135676211676816801144909383909637952422834338116302276833201292405930148853742835202417946949908422915255003078682769062345640307842830377701674520005088081945694660657027715635201020649655975823646472704126516550043038351200012876903486950091781192307958054051461722854455481599954751342918072942294648605842880661064029105920300036947422694980403296043169346146891365805184418207280463872
Technical details
Node id
n-6474462cc289ce0a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-427bf217233f512e

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:             361,             1064,              441;
  1:            6859,            30324,            25137,              5488;
  2:          130321,           768208,           955206,            417088,             60025;
  3:         2476099,         18244940,         30248190,          19811680,           5702375,            605052;
  4:        47045881,        415984632,        862073415,         752843840,         325035375,          68975928,          5764801;
  5:       893871739,       9220992676,      22931152839,       25032057680,       14409901625,        4586899212,        766718533,         52706752;
  6:     16983563041,     200227269536,     580922538588,      760974553472,      547576261750,      232402893408,      58270608508,       8011426304,       466948881;
  7:    322687697779,    4279857886332,   14191107728364,    21687774773952,    18727108151850,     9935223693192,    3321424684956,     684976948992,     79848258651,     4035360700;
  8:   6131066257801,   90352555378120,  337038808548645,   588668172435840,   593025091475250,   377538500341296,  157767672535410,   43381873436160,   7585584571845,   766718533000,   34179505129;
  9: 116490258898219, 1888368407402708, 7826790109629645, 15378956004886320, 17706034874046750, 13150924428555144, 6594688711980138, 2266702887039360, 528462391838535, 80122086698500, 7143516571961, 284735050992;
%F Formula
The row polynomials satisfy (361+(399*n+665)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (6859+(7581*n+22743)*x+(931*n^2+8379*n+15827)*x^2+(343*n^2+2058*n+3087)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 1064*x + 441*x^2.
%t Mathematica
P[0, x_] := 361 + 1064*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6859 + (7581*n + 22743)*x + (931*n^2 + 8379*n + 15827)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*P[n - 1, x])/(361 + (399*n + 665)*x + (49*n^2 + 196*n + 196)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 361 + 1064*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6859 + (7581*n + 22743)*x + (931*n^2 + 8379*n + 15827)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*v[i - 1])/(361 + (399*n + 665)*x + (49*n^2 + 196*n + 196)*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

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.