Real-rooted atlas

A038319 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(31 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 961 + 930\,x + 225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
09619302252116
1297914324520925337597336
292352117874601297350418500506254477456
3286291516926407567029750324337507846875759375205962976
4887503681257662359031168833752010892500729759375141243750113906259474296896
5275126141119318788650513527273847510909091812552785928125153249468752471765625170859375435817657216
68528910374413301513693320559127319030054109095390003272727543750126686227500030649893750042373125000256289062520047612231936
7264396221606711151402900545352228521742991002516072935635001826181969412508836364368125028504401187500591105093750071504648437538443359375922190162669056
881962828698080139659433241006508635521754090125111426087149550009435273508631250547854590823750022090910920312506108085968750001108322050781251191744140625057665039062542420747482776576
9254084768964048311352386673518321653271903242383036254749536964749568754596326094918937503113640257848312501506600124765312505207143288359375012597927310546875203192375976562519663778320312586497558593751951354384207722496
1078766278378854976145735258413528695801217148006166489485019631419454298217500213729163413730593751654677394170817500093409207735449375003874114606539375000117160723988085937525195854621093750036574627675781250321770917968750012974633789062589762301673555234816
1124417546297445042591153594242838767203395445918769531904784050791146204008218165250957031698397038103125833543737313549315625537770153105515687500260211364405894687500944315435343972656252538482353075195312549131916511132812506483683997070312505228777416992187519461950683593754129065876983540801536
12756943935220796320321512768472246345894411016127395498070556356475312143138672333348835004153517571043145367562540195331332675600331250291740308059742260468751613310459316547062500068305483156547355468752203402682469269531250533081294145791015625937972951576171875001134644699487304687584464865966796875029192926025390625189937030341242876870656
1323465261991844685929951170312385424679172072225576864531277139131212375120955466235529172673562517558051550318750871968751869082906969415415403125150732492497533501242187593773670447774298007812545374356668277886132812517076370789136838867187549576560355558564453125109039355620729980468751758699284205322265625196380813372802734375135747106018066406254378938903808593758737103395697172336050176
1472742312174718526382848156316628780427246231882402043748625096150064866700046149162502171130496990000725732797413175036041375008427864744153000418545000074763316278776616616125000516797117134400575687500002813210113433228940234375012099828444874102968750000409832898939284132812500010816704077576414062500002180787112414599609375003246829447763671875000033665282292480468750002171953696289062500006568408355712890625401906756202069927458308096
Technical details
Node id
n-9590dc4c111336bf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c2f5c2e31c1132c5

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:               961,                930,                225;
  1:             29791,              43245,              20925,               3375;
  2:            923521,            1787460,            1297350,             418500,              50625;
  3:          28629151,           69264075,           67029750,           32433750,            7846875,             759375;
  4:         887503681,         2576623590,         3116883375,         2010892500,          729759375,          141243750,           11390625;
  5:       27512614111,        93187886505,       135272738475,       109090918125,        52785928125,        15324946875,         2471765625,         170859375;
  6:      852891037441,      3301513693320,      5591273190300,      5410909539000,      3272727543750,      1266862275000,       306498937500,       42373125000,        2562890625;
  7:    26439622160671,    115140290054535,    222852174299100,    251607293563500,    182618196941250,     88363643681250,     28504401187500,     5911050937500,      715046484375,      38443359375;
  8:   819628286980801,   3965943324100650,   8635521754090125,  11142608714955000,   9435273508631250,   5478545908237500,   2209091092031250,   610808596875000,   110832205078125,   11917441406250,    576650390625;
  9: 25408476896404831, 135238667351832165, 327190324238303625, 474953696474956875, 459632609491893750, 311364025784831250, 150660012476531250, 52071432883593750, 12597927310546875, 2031923759765625, 196637783203125, 8649755859375;
%F Formula
The row polynomials satisfy P(n,x) = (31+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 961 + 930*x + 225*x^2.
%t Mathematica
P[0, x_] := 961 + 930*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(31 + 15*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 961 + 930*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (31 + 15*x)*v[i - 1]); 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.