Real-rooted atlas

A038321 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(324 + 630\,x + 196\,x^{2} + 378\,n x + 196\,n x^{2} + 49\,n^{2} x^{2}\right)\,P_{n}(x) = \left(5832 + 20412\,x + 14994\,x^{2} + 3087\,x^{3} + 6804\,n x + 7938\,n x^{2} + 2058\,n x^{3} + 882\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 1008\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
032410084411773
158322721623814548862350
2104976653184857304395136600252070625
31889568146966402571912017781120540225060505266093750
4340122243174474246944162406401203202917215006534561657648012048828125
5612220032666639590417499289248201637900801225230300041167738087263649265270675262089843750
6110199605761371372871684199829419525807171543044410829080001976051427845229827467275897722884669488811847900390625
71983592903682777030065152971960522803215679363166208142910862192008003008282752282410683228861477155532875645718722403536070054187011718750
8357046722662455540601303040218691117630720403183624273920428732586576000288108298179072127084807452960368862933196806808114684980726364926000341795051291569366455078125
9642684100792321099703905800192481120458787584099787947007795201212700744886400095075738399093765032558375137216182587151932416044933556920868071910127674000676754201554228473505099244975280761718750
10115683138142617621594185786621952103922019098118144239491072818708480327429201119328000293376564202917888181172101504939776788776496348037122426412073726872051775291925280007308945376785366150277101427223391775369691277256011962890625
112082296486567116842108662283912806422107047699054223365604091103957778432851315922910252800085812645029353482246056324536022272512307622833575734476811355608505041760963028854577628880005700977393892580871958242086698245473675436507461899024603977235986900329589843750
12374813367582081024816260222734309785646424800168013869056128384632563396378624214531612573383705600240275406082189750272190774222884701584128110744220087264411648476935557211753960321526542707124955520035916157581523259046044492335282652166896831049999399647855420020225441526001913910251006901264190673828125
13674664061647745843215742161438447403008096420738810490343424028886542326764185190405265775944983054592000648743596421912325734457232266865410475238403737617427945173893120183960857781676527552068694421820622998400019394725094022559881640800323263157902080620714794499945964064604817027304344041202051675576750121537574654540827999281883239746093750
14121439531096594251776302249499618190137753619835123412443727790080639948014623698871910401263786226795933102080001698528688813734089195521648289285723821686865921196037576942455645798406622590880140354991872028262847834770605056000930946804513082874316823500986199578951598084469146520399610940806202062434621217024059330954412830520003500282150050775049604316934614689774301588535308837890625
Technical details
Node id
n-995a122d765e012a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fc8dc5929055a459

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:            324,             1008,              441;
  1:           5832,            27216,            23814,             5488;
  2:         104976,           653184,           857304,           395136,             60025;
  3:        1889568,         14696640,         25719120,         17781120,           5402250,           605052;
  4:       34012224,        317447424,        694416240,        640120320,         291721500,         65345616,          5764801;
  5:      612220032,       6666395904,      17499289248,      20163790080,       12252303000,       4116773808,        726364926,         52706752;
  6:    11019960576,     137137287168,     419982941952,     580717154304,      441082908000,     197605142784,      52298274672,       7589772288,       466948881;
  7:   198359290368,    2777030065152,    9719605228032,   15679363166208,    14291086219200,    8003008282752,    2824106832288,     614771555328,     75645718722,     4035360700;
  8:  3570467226624,   55540601303040,  218691117630720,  403183624273920,   428732586576000,  288108298179072,  127084807452960,   36886293319680,   6808114684980,   726364926000,   34179505129;
  9: 64268410079232, 1099703905800192, 4811204587875840, 9978794700779520, 12127007448864000, 9507573839909376, 5032558375137216, 1825871519324160, 449335569208680, 71910127674000, 6767542015542, 284735050992;
%F Formula
The row polynomials satisfy (324+(378*n+630)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (5832+(6804*n+20412)*x+(882*n^2+7938*n+14994)*x^2+(343*n^2+2058*n+3087)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1008*x + 441*x^2.
%t Mathematica
P[0, x_] := 324 + 1008*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((5832 + (6804*n + 20412)*x + (882*n^2 + 7938*n + 14994)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*P[n - 1, x])/(324 + (378*n + 630)*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] = 324 + 1008*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((5832 + (6804*n + 20412)*x + (882*n^2 + 7938*n + 14994)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*v[i - 1])/(324 + (378*n + 630)*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.