Real-rooted atlas

A085880 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4\,n + 3\,n x + 3\,n^{2}\right)\,P_{n}(x) = \left(12 - 18\,x - 54\,x^{2} - 44\,n - 42\,n x + 36\,n x^{2} + 24\,n^{2} + 36\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2538
3326018110
42207206751351750
515687392120967560113429750
611424705601663201814408505010206525000
784480646272199584031363202566080962280962289487500
863148857657602205403245405360535134603502699210945935938223174281250
947590405051904023063040058810752090810720085621536046702656012509640093822303239843750
103609548843687987223188239367057290240134970675841667284819213100095008612471974414354811909569874660775000000
112751856643740805120226383206408010482688018284797440027975740083228798555968019394945856079342960320165297834009917870041148164062500
122106890240317839441922160314956808715753446402313026876160422378820864053853299660164751761734720280014530796010182346574401909189982701041376354221815117187500
131618832588826838540288020243927531529173029662720277563225139205892880779878489674272737280980011694914567566266762208039632825897280129707430209282210922105840110546105292416430273437500
141247596544002254802534400186911262720009398966353920031941799718400077321184145920013679901810432001784335018752000170627036168160011709698560560005520286464264001642399278624002566248872850011844225567007980371093750000
15963996549120188636597452801704630716006409420327641088003552809281781760965920023484416019484938404771840295485879105331203372393185441280028665342076250880177050642235667207587884667242880206942309106624029847448428840012791763612360153440771484375000
Technical details
Node id
n-8869c075c7629c90 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        5,         3;
   3:       32,        60,         18;
   4:      220,       720,        675,        135;
   5:     1568,      7392,      12096,       7560,        1134;
   6:    11424,     70560,     166320,     181440,       85050,       10206;
   7:    84480,    646272,    1995840,    3136320,     2566080,      962280,       96228;
   8:   631488,   5765760,   22054032,   45405360,    53513460,    35026992,    10945935,     938223;
   9:  4759040,  50519040,  230630400,  588107520,   908107200,   856215360,   467026560,  125096400,    9382230;
  10: 36095488, 436879872, 2318823936, 7057290240, 13497067584, 16672848192, 13100095008, 6124719744, 1435481190, 95698746;
%F Formula
The row polynomials satisfy (3*n^2-4*n+3*n*x)*P(n,x) = (24*n^2-44*n+12+(36*n^2-42*n-18)*x+(36*n-54)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((24*n^2 - 44*n + 12 + (36*n^2 - 42*n - 18)*x + (36*n - 54)*x^2)*P[n - 1, x])/(3*n^2 - 4*n + 3*n*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((24*n^2 - 44*n + 12 + (36*n^2 - 42*n - 18)*x + (36*n - 54)*x^2)*v[i - 1])/(3*n^2 - 4*n + 3*n*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.