Real-rooted atlas

A038329 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 + 20\,x + 4\,x^{2} + 12\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(192 + 432\,x + 204\,x^{2} + 27\,x^{3} + 144\,n x + 108\,n x^{2} + 18\,n x^{3} + 12\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 288\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
014428881513
117285184291643210260
2207368294469984207362025196425
32488321244160139968062208012150087483645000
4298598417915904251942401492992043740006298563572166065625
5358318082508226564232632323135283201224720002645395230005641399681175512500
642998169634398535686772211712601974374429393280008465264641440270721343692853144120605640625
751597803524643802316810448555212810835538739263489484800228562145285184974592725594112573956281968300356754375000
86191736422461917364224015672832819201857520926720126978969600054854914867215554923776029023764480344373768023619600071449296112494140625
974300837068881730920775682298682146816030649095290880239446056960001206808127078441064998768649577842278401515244579201558893600094313062825509168103797070312500
10891610044825610699320537907233101022914150449038552465408043100290252800024825767185612898555997044736275841857617925454880485120748268928000679054052163673320192898135291748980634765625
111069932053790721390911669927936469432688600678476500141846036487470716977152000484102460119449621963907912826887171888298065921701922711357442918248819200035310810712322865189749761401091052431248730829270773828125000
12128391846454886417974858503684096657205764040949761168365802739466241255080452161536009036579255563059246124206616936448172125319153582084765383591800832980531603251200148305404991744160450625986561176916484016524978677441076168025486938006103515625
131540702157458636823110532361879552090997721174900736017525487041091993602053768012627968000162658426600135065692248413233872896038728196809555968012253843521773568029415948097536000533899457970278472202781693952070614989040960472480809696019371024450036733201928057247583007812500
1418488425889503641629581481423205826561247968747541495808025883796245305098240328602882020474880002839128900656902963217711695340903596032826201531937193984029409224452256563208068374335324160001708478265504890882772586817047756833895194739660803023877182054401859618347200070527747686412440502369132681848785400390625
Technical details
Node id
n-f7b66e37e15b5e24 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9e3a89ee8f65d0f5

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:          144,           288,             81;
  1:         1728,          5184,           2916,            432;
  2:        20736,         82944,          69984,          20736,           2025;
  3:       248832,       1244160,        1399680,         622080,         121500,           8748;
  4:      2985984,      17915904,       25194240,       14929920,        4374000,         629856,         35721;
  5:     35831808,     250822656,      423263232,      313528320,      122472000,       26453952,       3000564,       139968;
  6:    429981696,    3439853568,     6772211712,     6019743744,     2939328000,      846526464,     144027072,     13436928,       531441;
  7:   5159780352,   46438023168,   104485552128,   108355387392,    63489484800,    22856214528,    5184974592,    725594112,     57395628,     1968300;
  8:  61917364224,  619173642240,  1567283281920,  1857520926720,  1269789696000,   548549148672,  155549237760,  29023764480,   3443737680,   236196000,   7144929;
  9: 743008370688, 8173092077568, 22986821468160, 30649095290880, 23944605696000, 12068081270784, 4106499876864, 957784227840, 151524457920, 15588936000, 943130628, 25509168;
%F Formula
The row polynomials satisfy (16+(12*n+20)*x+(n^2+4*n+4)*x^2)*P(n,x) = (192+(144*n+432)*x+(12*n^2+108*n+204)*x^2+(3*n^2+18*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 288*x + 81*x^2.
%t Mathematica
P[0, x_] := 144 + 288*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((192 + (144*n + 432)*x + (12*n^2 + 108*n + 204)*x^2 + (3*n^2 + 18*n + 27)*x^3)*P[n - 1, x])/(16 + (12*n + 20)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 288*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((192 + (144*n + 432)*x + (12*n^2 + 108*n + 204)*x^2 + (3*n^2 + 18*n + 27)*x^3)*v[i - 1])/(16 + (12*n + 20)*x + (n^2 + 4*n + 4)*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.