Real-rooted atlas

A034870 → D → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 12\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27Pn(1)
00
14610
232144216108500
31921440432064804860145818750
4102410752483841209601814401632968164817496625000
551206912041472014515203265920489888048988803149280118098019683019531250
624576405504304128013685760410572808622028812933043213856832010392624051963120155889362125764585937500
71146882236416201277441107025924151347201120863744224172748833625912323782915136315242928018914575687737780961934445242232052217089843750
852428811796480123863040805109760362299392011955879936298896998405764442112086466631680100877736960907899632646190224768030951123840107138505602295825120229582512488281250000
923592966016204872194457654145843202842656768011086361395233259084185678396412723214699327385602204899107840264587892940825256117053441894208779008109281275712046834832448014050449734426344593252232452293413732910156250
10104857602988441604034396160342923673602057542041609258939187203240628715520902746570752020311797841920372382960435205585744406528068552317716480685523177164805536917969408035594472660480177972363302406673963623840176663742984029443957164023245229340381469726562500
1146137344145332633621799895040207099002880139791826944071293831741442851753269657691663497953280240616682127360521336144609280938405060296704140760759044505617595094880563201827182929904640156615679706112010963097579427846166742388428162720621641953609068738806512021478591910160322178878652423012777046610490417480468750
12201326592694576742411460516249612033542062089025156546560514433923153922314952654192648433041811701762529912543510528632478135877632013282040853430272235454360583536643531815408753049644826887880327168480288084432076804322592759888691232419445699165184200237752847784961001188764238924839520609114694401185618273440832254061058594464346446898083362259436291848286102294921875000
138724152323271557120058888028160067721232384005587001671680035198110531584017599055265792007165329643929600241829875482624006851846472007680016444431532818432033636337226219520058863590145884160088295385218826240011352263813849088001248749019523399680117070220580318720092967528107900160061978352071933440034251194566068480015413037554730816055046562695467200150126989169456002937267179402400367158397425300220295038455187748603820800781250
14375809638415220290355229679566192643709945774080033389511966720023038763257036801267131979137024057020939061166080213828521479372800677123651351347200182823385864863744042381784905036595208476356981007319040146706178517434368002200592677761515520028607704810899701760321836679122621644803123708944425445376026030907870212044800184956450656769792001109738703940618752055486935197030937602269920076242174720740191329209404800185047832302351200333086098144232163843301132433448213516729579636208616256713867187500
Technical details
Node id
n-c9845261e64d60a9 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-484d946563c33157

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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:       0;
  1:       4,        6;
  2:      32,      144,       216,        108;
  3:     192,     1440,      4320,       6480,        4860,         1458;
  4:    1024,    10752,     48384,     120960,      181440,       163296,        81648,        17496;
  5:    5120,    69120,    414720,    1451520,     3265920,      4898880,      4898880,      3149280,       1180980,        196830;
  6:   24576,   405504,   3041280,   13685760,    41057280,     86220288,    129330432,    138568320,     103926240,      51963120,      15588936,       2125764;
  7:  114688,  2236416,  20127744,  110702592,   415134720,   1120863744,   2241727488,   3362591232,    3782915136,    3152429280,    1891457568,     773778096,     193444524,      22320522;
  8:  524288, 11796480, 123863040,  805109760,  3622993920,  11955879936,  29889699840,  57644421120,   86466631680,  100877736960,   90789963264,   61902247680,   30951123840,   10713850560,   2295825120,    229582512;
  9: 2359296, 60162048, 721944576, 5414584320, 28426567680, 110863613952, 332590841856, 783964127232, 1469932738560, 2204899107840, 2645878929408, 2525611705344, 1894208779008, 1092812757120, 468348324480, 140504497344, 26344593252, 2324522934;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+12*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 6*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 12*n*x + 9*n*x^2)*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] = 0; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 12*n*x + 9*n*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.