Real-rooted atlas

A038226 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(22 + 32\,x + 11\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 352 + 512\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0352512864
15808168961228834992
2851843717125406722621441259712
3117128068147201486848014417920524288042515280
4154608961124428803271065604757913603460300801006632961377495072
51984148321731620352629680128012211978240133221580807751073792187904819243391094768
6249435788825397098496110823702528268663521280390783303680341047246848165356240896343597383681338925209984
730867678864359187535872182859109171253195377213449671886766080112545591459848185133924352340161409843261847529062440669853253264
83772716305604938828618240287350028697609752485822464021278150885376030950037651456030012157722624018708877541376068032281968640109951162777601220095597597920
9456498672977666399806978560434616918405120168578683502592042910937618841607489909111652352907867771109376075459139416883204115953059102720133040906960896019351404648857636236839248658224
105477984075731287647745211699263743814699417602781548277792768080917768081244160164778000456351744239677091572875264249015160075714560181101934600519680878069985941913602554385413649203233776997205278721067339628778660416
11652793102357968113942068775208969115365502016716844195711524929536014464051044522393603366179152179757056571230401582019379271218335781654364166474394161968578560418546693298978816018263855707591802884830110600354856965854679515581644831219684141775817168
127733087520240544146225654928184832127615117028234035268061395748391485442474959845396054016064798948679460323328125670688348044263424182793728506246201344199411340188632219648161140476920106844160937544592989712547843719185162273239859290162064539957329921008806316530991104907772354276250683808
139113996005997784018559410048577305601754707859138217984010209209362258722816040836837449034891264011879807257901059276802591957947178412933120430870931479008903168054838118551873860403205317635738363525857280386737144608256425984020455518392502819225607438370324546479718401664530422276135321601729382256910270464026260557391562966210160
14106937553137040665623331829775354327040237560448621789511680149735070646461267968065338939918455826022402090846077390586432716850687177633711186247680947916049253819586969601378786980732828490137601559839816586634251468801361314749021062619463689000428092701240459264043638439237339347681280146478677160299908300803043712772162076016640295147905179352825856756304052877013426852608
Technical details
Node id
n-d0bed16060b50d92 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-89a932e93fe55d02

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:           352,            512;
  1:          5808,          16896,           12288;
  2:         85184,         371712,          540672,           262144;
  3:       1171280,        6814720,        14868480,         14417920,          5242880;
  4:      15460896,      112442880,       327106560,        475791360,        346030080,        100663296;
  5:     198414832,     1731620352,      6296801280,      12211978240,      13322158080,       7751073792,       1879048192;
  6:    2494357888,    25397098496,    110823702528,     268663521280,     390783303680,     341047246848,     165356240896,      34359738368;
  7:   30867678864,   359187535872,   1828591091712,    5319537721344,    9671886766080,   11254559145984,    8185133924352,    3401614098432,     618475290624;
  8:  377271630560,  4938828618240,  28735002869760,   97524858224640,  212781508853760,  309500376514560,  300121577226240,  187088775413760,   68032281968640,   10995116277760;
  9: 4564986729776, 66399806978560, 434616918405120, 1685786835025920, 4291093761884160, 7489909111652352, 9078677711093760, 7545913941688320, 4115953059102720, 1330409069608960, 193514046488576;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(16*n+32)*x)*P(n-1,x) for n >= 1, with P(0,x) = 352 + 512*x.
%t Mathematica
P[0, x_] := 352 + 512*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (16*n + 32)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 352 + 512*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (16*n + 32)*x)*v[i - 1])/(n + 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

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.