Real-rooted atlas

A038281 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 + 120\,x + 36\,x^{2} + 72\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(512 + 1728\,x + 1224\,x^{2} + 243\,x^{3} + 576\,n x + 648\,n x^{2} + 162\,n x^{3} + 72\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 64 + 192\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
06419281337
1512230419444325192
24096245763110413824202575625
3327682457604147202764808100087481059476
426214423592964976640442368019440004199043572114421385
5209715222020096557383686193152036288000117573122000376139968191972792
6167772162013265925945425927927234565806080002508226566401203289579525314412510301937
713421772818119393286115295232951268147283607552004514807808153628876832248627238263752196830032348703860
8107374182416106127360611529523201087163596801114767360007223692492830725775360859963392015305500801574640007144929411783410401
98589934592141733920768597939978240119587995648014014218240001059474898944540773646336189191946240448961356806928416000628753752255091685187484920200
1068719476736123695058124857402237911041275605286912016817061888000145299414712328652378341376363248536780810775072563202217093120003018018009624488801288981352964765749228697
1154975581388810720238370816542712067522561326629498388481943304929280001888892391260161285496210718726296307970867222412150931456576444211200010462462433281273417666569340607016312487308802296417758132
12439804651110492358976733184506531263021056135075003472281621765015207936002350621642457088179969469500620810074092753387524183601507205121291235033088002929489481318447540926218245230739928963499857849610761680259870357244787545
133518437208883279164837199872046756731971174401350750034722816023743652954112000282074597094850562399592926674944015111139130081280717188829806592025824700661760007030774755164161426227786547202092295971584020999147097601291401630003673320192120691401655182776
1428147497671065667553994410557444274901208793088013299692649578496025326563151052800032823225843764428830714789461439283221491397873893376011475021276905472047222309781504000149989861443502083651143133560832669534710906880895963609497608264970432000470184984576124405023691467733106799226465
Technical details
Node id
n-8b2de826e1bf0a0d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-77376c12e17ddecb

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:         64,          192,           81;
  1:        512,         2304,         1944,           432;
  2:       4096,        24576,        31104,         13824,          2025;
  3:      32768,       245760,       414720,        276480,         81000,          8748;
  4:     262144,      2359296,      4976640,       4423680,       1944000,        419904,        35721;
  5:    2097152,     22020096,     55738368,      61931520,      36288000,      11757312,      2000376,       139968;
  6:   16777216,    201326592,    594542592,     792723456,     580608000,     250822656,     64012032,      8957952,      531441;
  7:  134217728,   1811939328,   6115295232,    9512681472,    8360755200,    4514807808,   1536288768,    322486272,    38263752,    1968300;
  8: 1073741824,  16106127360,  61152952320,  108716359680,  111476736000,   72236924928,  30725775360,   8599633920,  1530550080,  157464000,   7144929;
  9: 8589934592, 141733920768, 597939978240, 1195879956480, 1401421824000, 1059474898944, 540773646336, 189191946240, 44896135680, 6928416000, 628753752, 25509168;
%F Formula
The row polynomials satisfy (64+(72*n+120)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (512+(576*n+1728)*x+(72*n^2+648*n+1224)*x^2+(27*n^2+162*n+243)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 192*x + 81*x^2.
%t Mathematica
P[0, x_] := 64 + 192*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((512 + (576*n + 1728)*x + (72*n^2 + 648*n + 1224)*x^2 + (27*n^2 + 162*n + 243)*x^3)*P[n - 1, x])/(64 + (72*n + 120)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 64 + 192*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((512 + (576*n + 1728)*x + (72*n^2 + 648*n + 1224)*x^2 + (27*n^2 + 162*n + 243)*x^3)*v[i - 1])/(64 + (72*n + 120)*x + (9*n^2 + 36*n + 36)*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.