Real-rooted atlas

A038281 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 660\,x + 576\,x^{2} + 396\,n x + 576\,n x^{2} + 144\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 13068\,x + 26928\,x^{2} + 15552\,x^{3} + 4356\,n x + 14256\,n x^{2} + 10368\,n x^{3} + 1584\,n^{2} x^{2} + 1728\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 1056\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121105612962473
1133117424427682764889171
21464125555294089612165125184002946001
31610513513840172497603345408028512000895795291848683
41771561463826882846210407359897609408960005912248321463132162747199097
519487171595244496438316401614167802880241496640002276215603211266117632229323571279636871939
6214358881748307366464286405568249353330688531292608000667689910272495709175808201804742656348285173762252662122913
723579476919260303659290919345017641143299563521051959363840016525325279232163584028016649989334761472344802322022451597803520062475142127003
825937424601113181489168012501409939920646537564569601928592167040003635571561431044498560770457603662756079206401896412771123205675758387200074920010711041704751838582089
9285311670611136949601893281680745114144809778880664115203333709317312000733173598221926410886517064507392110798371395993607648864843530240343383382425600090653212960358410699320537907245887966356092851
10313842837672116433952227193622185835506711361434235830736896055006203735648000138255592807563264239503375419162624292507700485423104252412539836497920151088688267264000598311205538365441412310311003750415068209757552641220963564999876977
113452271214393119583793070739042884158615872476820509572379537612887398745935529600024713187214351933444892711812134322176697143352823591731272189986393238405125401420605554688000285195007973287526410098018723676815362154753995330027522097066825429811232163998997900433739
1237974983358324123199262560721632370133689036967856287134013313526579213459406874071558400422870092334466416649418470238358570188815337153762119018086418528763174264523980816636375465108439040010980007806971569766451836496114874318848165916057640412119043229482911161909248288881654523494400839965639854906883609
13417724816941565127341988017993352046978506685461304803948092683060990464020189110311107337600069773565235186958745617267195436990712012803163287963437047480320436749417679092350976045750032529048207360003623402576300618022912213825546473856565248091253831702226665472026643234017085751296047665472996376576000394419752309411020821768826435674089804387
1445949729863572161320812659411121996859058694118865640320534511009399026401280296106951229574284800011163770437629913399296303902639691036531425286185985350721337294848096084871889400317214720115028653215892635648000106286475571484795338752752665923587975109672964015168594897973280768015630697290023640760320419456162368113868800069417876406456339660853431550820665524224560362929419229549517633
Technical details
Node id
n-b5ec2b0228e31a42 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1bffe7ba5171bd1a

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:          121,           1056,            1296;
  1:         1331,          17424,           42768,           27648;
  2:        14641,         255552,          940896,         1216512,           518400;
  3:       161051,        3513840,        17249760,        33454080,         28512000,          8957952;
  4:      1771561,       46382688,       284621040,       735989760,        940896000,        591224832,         146313216;
  5:     19487171,      595244496,      4383164016,     14167802880,      24149664000,      22762156032,       11266117632,        2293235712;
  6:    214358881,     7483073664,     64286405568,    249353330688,     531292608000,     667689910272,      495709175808,      201804742656,      34828517376;
  7:   2357947691,    92603036592,    909193450176,   4114329956352,   10519593638400,   16525325279232,    16358402801664,     9989334761472,    3448023220224,     515978035200;
  8:  25937424601,  1131814891680,  12501409939920,  64653756456960,  192859216704000,  363557156143104,   449856077045760,   366275607920640,  189641277112320,   56757583872000,   7492001071104;
  9: 285311670611, 13694960189328, 168074511414480, 977888066411520, 3333709317312000, 7331735982219264, 10886517064507392, 11079837139599360, 7648864843530240, 3433833824256000, 906532129603584, 106993205379072;
%F Formula
The row polynomials satisfy (121+(396*n+660)*x+(144*n^2+576*n+576)*x^2)*P(n,x) = (1331+(4356*n+13068)*x+(1584*n^2+14256*n+26928)*x^2+(1728*n^2+10368*n+15552)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 1056*x + 1296*x^2.
%t Mathematica
P[0, x_] := 121 + 1056*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (4356*n + 13068)*x + (1584*n^2 + 14256*n + 26928)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*P[n - 1, x])/(121 + (396*n + 660)*x + (144*n^2 + 576*n + 576)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 1056*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (4356*n + 13068)*x + (1584*n^2 + 14256*n + 26928)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*v[i - 1])/(121 + (396*n + 660)*x + (144*n^2 + 576*n + 576)*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

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.