Real-rooted atlas

A038321 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 385\,x + 196\,x^{2} + 231\,n x + 196\,n x^{2} + 49\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 7623\,x + 9163\,x^{2} + 3087\,x^{3} + 2541\,n x + 4851\,n x^{2} + 2058\,n x^{3} + 539\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 616\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01216164411178
113311016414553548831536
21464114907232016624147260025785376
3161051204974058697106640480330137560505218627408
417715612705656896850215146090560108945375399334325764801426412512
5194871713472259561491493311281224328027962646251537437132443889677527067529500747904
6214358881436512630421875235228494954817286151782175045098155872195311457884638194176466948881207202468608
7235794769154018438012309378326796816675448512121805287065011161793578326445278110042295906117124622793921940353607004441044112128
825937424601660225353480425395199344512833471333760223309692952502455594587230417724514802610841832242944025425366570454438896770003417950512993823944344064
92853116706117988726777108571920212452051941062539231203860067549607504952115750914644289332582231622546542534905601025489785008152685532545850041357201206092847350509921958202914512896
10313842837672195864721325296754934680436706284689172420576063691114568523759338275416010464943653168090956467228722921507843384116290526895118163432017400027295752796019437585026730944233917753696940446252743196672
11345227121439311142387929126444981415084567717840710551656142368101198104258876625166921673061187044192774861481438236160228456296260352967857259090691974224342694622050013010975499435914268732941126249633450238778656718990246039772827905658640666624
123797498335832411353290316042095212594826918619045156994772318599315215584508055867000252856215294602533864371091608351768604335250260385177277442484166964999442723130109754994359140050092255672828268913794957644480812825756683859565659292449789012488815260019139102516813387297648041984
134177248169415651159494930104961220159857418582472495578367811938074058402337676208380050037547127552360941808756680334615311575774557270366204442813472058555364174986864185357801826234487635001653044437203332873756904200283483352801416617612276111245241271075935303260251790315795191251215375746545408339081159872589299712
14459497298635721611871407179898211648200963611932251137201060979607777002636803428591772290740055007540408377750689400961197388922948373363208142176050220215019008012882180118497110120708996160202467117680004848930349129776429522003027849978614018566233117494014889478014154569788204457920221575477899768300021390613139199180896043169346146896796115980998293127168
Technical details
Node id
n-af37e30b10569b18 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7c6977712369e032

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:          121,           616,            441;
  1:         1331,         10164,          14553,            5488;
  2:        14641,        149072,         320166,          241472,           60025;
  3:       161051,       2049740,        5869710,         6640480,         3301375,          605052;
  4:      1771561,      27056568,       96850215,       146090560,       108945375,        39933432,         5764801;
  5:     19487171,     347225956,     1491493311,      2812243280,      2796264625,      1537437132,       443889677,        52706752;
  6:    214358881,    4365126304,    21875235228,     49495481728,     61517821750,     45098155872,     19531145788,      4638194176,       466948881;
  7:   2357947691,   54018438012,   309378326796,    816675448512,   1218052870650,   1116179357832,    644527811004,    229590611712,     46227939219,     4035360700;
  8:  25937424601,  660225353480,  4253951993445,  12833471333760,  22330969295250,  24555945872304,  17724514802610,   8418322429440,   2542536657045,   443889677000,   34179505129;
  9: 285311670611, 7988726777108, 57192021245205, 194106253923120, 386006754960750, 495211575091464, 428933258223162, 254654253490560, 102548978500815, 26855325458500, 4135720120609, 284735050992;
%F Formula
The row polynomials satisfy (121+(231*n+385)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (1331+(2541*n+7623)*x+(539*n^2+4851*n+9163)*x^2+(343*n^2+2058*n+3087)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 616*x + 441*x^2.
%t Mathematica
P[0, x_] := 121 + 616*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (2541*n + 7623)*x + (539*n^2 + 4851*n + 9163)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*P[n - 1, x])/(121 + (231*n + 385)*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] = 121 + 616*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (2541*n + 7623)*x + (539*n^2 + 4851*n + 9163)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*v[i - 1])/(121 + (231*n + 385)*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.