Real-rooted atlas

A109983 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 4\,x + 4\,x^{2} - n - 4\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 10\,x - 8\,x^{2} + 6\,n + 20\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 10\,x + 8\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1310821
2138822024096657
3636422700599273924800128022869
4321436825824865921800162374401939208960017920836001
516832865021828097984028688805722720787328073740804497920161280025804831436181
68989183720171332496383203641928097355520188743360267321600274444800199127040968970242838528037847041204022961
748639116019412800060865611604007930081343864480336448224063880864009241648640101358028808294221824490970726419869696004920115205622988846714387797
8265729724585692309920729237376399788569616126136320495012044801179397427202203917606403240456806403735837450243341098844162272052428801135562588163933280665684344832008434483201829897338689
91462563448749546482952725858890176371145300481749892838406363302400001824612710400418176494208077168328985601149193154764813776668319744131950128414729958952779776579229699276825047603609607582600396801433862144001274544128072212212665621
108097453276098040446001750045380669040326170155360176013895372873990220544002480799992832067374781320960149663960094720273390461854720411357404057600508893449011200514753698201600421538249932800275148867502080139782722027520532631991091201431950327808024216338432001937307074562866445116914897
1145046719168976269030180509964341423144040274513731888016685419041984796106091686403056093693706249597337768972802493441283294720539828461359616097786310436352001483929505529856018838790398771200199241094733824001742553953107968012460594068848640716390983925760032304276832256001100148254965760265992261337088406834485657602958796259328114327563401778517
1225159596910296750384201485367360250858784352022309512504048150815632581504805224736297344348264271795968012416825507201280369429996051578889250059572852736019599873943781376035254617780951040053883576533729280069903652002742272076725177971965952070845828190765056054564190027776000034622471223115776017783897752338432072087293109731328221914650776698884874616837242880680523139645440453682093096964578036254707037409
131409933619625166058661330036336200180729176434561761445268412961310914477508064774382091178009637259845588542720148689244268987136498523240335284736141739445108591616034398082996480819207156477219127808000127954806771310182401967617526790815744026000033914234798080294471007536486809602845904366236991488023314480272963010560160392578036504985609146167250626019328424522428590535475215627819395789619204389147409666539528833190352597811211342052327424000697972450918400183932435144138658453
14792384825337839902906486963677636921280639581908961357409629784208110282643353326087141237081876934437836058653422515216708760408874293766233398123166742528198398488243052697605426592452907644928012821393081062279168026258229829213290496046706498021811085312072200359026237145088096927294475038949376011277197712645423104001133091023071810682880978039572593524080640719941317612049268736447465953949781065728231671582499352346624980753843142703185923306666472358123929685382969920847872001585444422261145600188452561747968000107687178141696007411004320351849711857
Technical details
Node id
n-dbf83e3ea48cbbcf · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-1d4b84e819f54819

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:       1;
  1:       3,       10,         8;
  2:      13,       88,       220,        240,          96;
  3:      63,      642,      2700,       5992,        7392,         4800,         1280;
  4:     321,     4368,     25824,      86592,      180016,       237440,       193920,         89600,         17920;
  5:    1683,    28650,    218280,     979840,     2868880,      5722720,      7873280,       7374080,       4497920,       1612800,         258048;
  6:    8989,   183720,   1713324,    9638320,    36419280,     97355520,    188743360,     267321600,     274444800,     199127040,       96897024,       28385280,        3784704;
  7:   48639,  1160194,  12800060,   86561160,   400793008,   1343864480,   3364482240,    6388086400,    9241648640,   10135802880,     8294221824,     4909707264,     1986969600,     492011520,      56229888;
  8:  265729,  7245856,  92309920,  729237376,  3997885696,  16126136320,  49501204480,  117939742720,  220391760640,  324045680640,   373583745024,   334109884416,   227205242880,  113556258816,   39332806656,    8434483200,    843448320;
  9: 1462563, 44874954, 648295272, 5858890176, 37114530048, 174989283840, 636330240000, 1824612710400, 4181764942080, 7716832898560, 11491931547648, 13776668319744, 13195012841472, 9958952779776, 5792296992768, 2504760360960, 758260039680, 143386214400, 12745441280;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n-1+(4*n-4)*x+(4*n-4)*x^2)*P(n-2,x) + (6*n-3+(20*n-10)*x+(16*n-8)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 10*x + 8*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 10*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (4*n - 4)*x + (4*n - 4)*x^2)*P[n - 2, x] + (6*n - 3 + (20*n - 10)*x + (16*n - 8)*x^2)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 10*x + 8*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (4*n - 4)*x + (4*n - 4)*x^2)*v[i - 2] + (6*n - 3 + (20*n - 10)*x + (16*n - 8)*x^2)*v[i - 1])/(n)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.