Real-rooted atlas

A094638 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n x^{2} + 6\,n^{2} x + 18\,n^{2} x^{2} - n^{3} - 6\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 6\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1268
264254102
3242767836481731
41201956888315768972036447
57201533697308272268353970174960914562
650401332721095228424126886038208870472367416026623260
7403201277856129598926459652818016384528778004024671984488179840881718165
83628801343606416264184499608529635593704457778739096102783197167560791424238085568032730602445
9362880015395616021705790321581595934469211583190193110993378347813478684392767494048253467596304714256704001345940939340
1039916800191124576030804396552260798714064135442589049046158379219981058483303375416245475989408160440464834649234602251200235704712320060729822966690
1147900160025569457920464259423744448497801705627011057641020109025320327380304727756401971594145129145904795641853908340699096931408080363500627196960848536964352002982977658365175
1262270208003668910681607416123236352805891665614405529677601007802584206854348004854052428697558320261257228006416344062218312175324091393426463864032409746637320448153787590490291203309294160972800158445290480496075
138717829120056221825766401252888527962881513886740150464116702705112811206205778713358856023714875189049234466432096941157295213706497665158269502063731359057700680220894423287548392815941900290742231046960826880750995201389903547608576009049431094259941350
141307674368000916557151104002233530294272640297244768588490882545331115271161601519048440772396848659806284790322460021347954631917226792519489123024342262329491962615660410120012847855228159803033612512265892422642835282976886644003644032335692107512712115206254565964238592000553022063027038798200
Technical details
Node id
n-25aab4bf90388489 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1f228a2711e045ca

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2,         6;
  2:       6,        42,         54;
  3:      24,       276,        783,         648;
  4:     120,      1956,       8883,       15768,        9720;
  5:     720,     15336,      97308,      272268,      353970,       174960;
  6:    5040,    133272,    1095228,     4241268,     8603820,      8870472,      3674160;
  7:   40320,   1277856,   12959892,    64596528,   180163845,    287780040,    246719844,     88179840;
  8:  362880,  13436064,  162641844,   996085296,  3559370445,   7778739096,  10278319716,   7560791424,   2380855680;
  9: 3628800, 153956160, 2170579032, 15815959344, 69211583190, 193110993378, 347813478684, 392767494048, 253467596304, 71425670400;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(6*n^3-6*n^2)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (2*n^2-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(n - 1)); 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.