Real-rooted atlas

A145324 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 8\,x + 6\,x^{2} + n + 6\,n x + 5\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131114
21292176280
360712281236967280
43605832353849528896096232960
52520519844284801763920362660029789768852480
62016050659252992642953664092513920154389408107243136389509120
7181440538588868463648483099584204362752051825753927295153952439696857619475456000
81814400622051209323873287980323648426588560641458326198403113500075523795467675522022605544961090625536000
9199584007767907201342872172813533531385687624392275237797604775041086231369491220053864828224215817512446081031528827929667618783232000
102395008001043915904020464494105623760321830401809370464896094426785404160342014351079168848935944853760138199744531584013323615290496005776561436405764598077259776000
11311351040015031861632032971729351684337175978393638015612360192023312641938406401020622047368294431899042698928128697510594251852801016225440419110408878358313935308835237024762075136340257717223424000
124358914560023099523148805608144976947282481805069619281847218761666565772800287149568029675052342504345611201971490480680963081851650081850368602628553864898304079500580707170718726353034833866355712232564363429695897627220617377873920000
136538371840003774411406080010052283608985601635405369407539218133290974192947214470352962802677768549946055251845120378722443988991068161257617723336405903363092057502355461217285471171443048338739206597496455289074385924859501277189656401921651206980350840872962340973094497157120000
14104613949440006535974509568001895220644299776033806221453349683241442346366206013443693386588087092428824677381940132987289612557518105815259381764890478931654581563392145051867011254228131843225351132677884723404852136897295629893435392579161751037004155760643957414087520273405132812549173050666390634496215369524693738455040000
Technical details
Node id
n-2bf8705f608686c7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-295fd44668f606da

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        3,        11;
  2:       12,        92,         176;
  3:       60,       712,        2812,         3696;
  4:      360,      5832,       35384,        95288,        96096;
  5:     2520,     51984,      428480,      1763920,      3626600,       2978976;
  6:    20160,    506592,     5299264,     29536640,     92513920,     154389408,      107243136;
  7:   181440,   5385888,    68463648,    483099584,   2043627520,    5182575392,     7295153952,     4396968576;
  8:  1814400,  62205120,   932387328,   7980323648,  42658856064,  145832619840,   311350007552,   379546767552,   202260554496;
  9: 19958400, 776790720, 13428721728, 135335313856, 876243922752, 3779760477504, 10862313694912, 20053864828224, 21581751244608, 10315288279296;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+2+(6*n+8)*x+(5*n+6)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (6*n + 8)*x + (5*n + 6)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 2 + (6*n + 8)*x + (5*n + 6)*x^2)*v[i - 1])/(1 + x)); 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. [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.