Real-rooted atlas

A091042 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-48 + 198\,x - 27\,x^{2} + 128\,n - 762\,n x + 180\,n x^{2} - 112\,n^{2} + 1068\,n^{2} x - 459\,n^{2} x^{2} + 32\,n^{3} - 648\,n^{3} x + 558\,n^{3} x^{2} + 144\,n^{4} x - 324\,n^{4} x^{2} + 72\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-192 - 936\,x - 1188\,x^{2} - 324\,x^{3} + 512\,n + 1560\,n x - 72\,n x^{2} - 270\,n x^{3} - 448\,n^{2} + 240\,n^{2} x + 4896\,n^{2} x^{2} + 1701\,n^{2} x^{3} + 128\,n^{3} - 1440\,n^{3} x - 4824\,n^{3} x^{2} - 594\,n^{3} x^{3} + 576\,n^{4} x + 864\,n^{4} x^{2} - 1188\,n^{4} x^{3} + 288\,n^{5} x^{2} + 648\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-64\,x - 324\,x^{2} - 450\,x^{3} - 162\,x^{4} + 128\,n x + 360\,n x^{2} - 144\,n x^{3} - 216\,n x^{4} - 64\,n^{2} x + 240\,n^{2} x^{2} + 1548\,n^{2} x^{3} + 756\,n^{2} x^{4} - 288\,n^{3} x^{2} - 864\,n^{3} x^{3} - 144\,n^{4} x^{3} - 432\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
243640
316240405661
4641344453630248968
52566912349925184018225112225
6102433792228096532224445500962281336864
74096159744134784043130885896800318427246437315370213
8163847372807464960304128005832000052907904200037602099520171962608
96553633423363948134419552665648470400062817638440807670411421388890344971882621345
1026214414942208201719808117669888035852544006084688896573364915228083179526058427403739770020248773880
111048576660602881003290624673814937624385536000513559388166407604403246115536896176778534243031182000150043509214600683541
124194304289406976488374272037116444672155669299200392286633984612467122176589254082560335598614208103579819200144613362965867108642246197407160
13167772161258291200233570304001981808640009455616000002777681756160522338181120063350636544004863748032000225173520000057216591432066323836800224533822523260720105921
146710886454358179841100753141761031446462464551809843200018528771244032408319218155525999792593305658449870895104367483184640001412432428492830084492372482942291210048437157316238657371287728
152684354562335388467251232584499252546240512003115827855360011777829293260829796627913113651553773394329661169760423936049086844784640025789969898035283705987948544151691457939841268698470480312088727252428871948928805
Technical details
Node id
n-5aece5ec0befd6fe · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-44d9798d5385e5d1

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,       36;
   3:     16,      240,       405;
   4:     64,     1344,      4536,       3024;
   5:    256,     6912,     34992,      51840,      18225;
   6:   1024,    33792,    228096,     532224,     445500,      96228;
   7:   4096,   159744,   1347840,    4313088,    5896800,    3184272,     464373;
   8:  16384,   737280,   7464960,   30412800,   58320000,   52907904,   20003760,    2099520;
   9:  65536,  3342336,  39481344,  195526656,  484704000,  628176384,  408076704,  114213888,   9034497;
  10: 262144, 14942208, 201719808, 1176698880, 3585254400, 6084688896, 5733649152, 2808317952, 605842740, 37397700;
%F Formula
The row polynomials satisfy (32*n^3-112*n^2+128*n-48+(144*n^4-648*n^3+1068*n^2-762*n+198)*x+(72*n^5-324*n^4+558*n^3-459*n^2+180*n-27)*x^2)*P(n,x) = (128*n^3-448*n^2+512*n-192+(576*n^4-1440*n^3+240*n^2+1560*n-936)*x+(288*n^5+864*n^4-4824*n^3+4896*n^2-72*n-1188)*x^2+(648*n^5-1188*n^4-594*n^3+1701*n^2-270*n-324)*x^3)*P(n-1,x) - ((64*n^2-128*n+64)*x+(288*n^3-240*n^2-360*n+324)*x^2+(144*n^4+864*n^3-1548*n^2+144*n+450)*x^3+(432*n^4-756*n^2+216*n+162)*x^4)*d/dx 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[((128*n^3 - 448*n^2 + 512*n - 192 + (576*n^4 - 1440*n^3 + 240*n^2 + 1560*n - 936)*x + (288*n^5 + 864*n^4 - 4824*n^3 + 4896*n^2 - 72*n - 1188)*x^2 + (648*n^5 - 1188*n^4 - 594*n^3 + 1701*n^2 - 270*n - 324)*x^3)*P[n - 1, x] - ((64*n^2 - 128*n + 64)*x + (288*n^3 - 240*n^2 - 360*n + 324)*x^2 + (144*n^4 + 864*n^3 - 1548*n^2 + 144*n + 450)*x^3 + (432*n^4 - 756*n^2 + 216*n + 162)*x^4)*D[P[n - 1, x], x])/(32*n^3 - 112*n^2 + 128*n - 48 + (144*n^4 - 648*n^3 + 1068*n^2 - 762*n + 198)*x + (72*n^5 - 324*n^4 + 558*n^3 - 459*n^2 + 180*n - 27)*x^2)]];
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] = ((128*n^3 - 448*n^2 + 512*n - 192 + (576*n^4 - 1440*n^3 + 240*n^2 + 1560*n - 936)*x + (288*n^5 + 864*n^4 - 4824*n^3 + 4896*n^2 - 72*n - 1188)*x^2 + (648*n^5 - 1188*n^4 - 594*n^3 + 1701*n^2 - 270*n - 324)*x^3)*v[i - 1] - ((64*n^2 - 128*n + 64)*x + (288*n^3 - 240*n^2 - 360*n + 324)*x^2 + (144*n^4 + 864*n^3 - 1548*n^2 + 144*n + 450)*x^3 + (432*n^4 - 756*n^2 + 216*n + 162)*x^4)*deriv(v[i - 1], x))/(32*n^3 - 112*n^2 + 128*n - 48 + (144*n^4 - 648*n^3 + 1068*n^2 - 762*n + 198)*x + (72*n^5 - 324*n^4 + 558*n^3 - 459*n^2 + 180*n - 27)*x^2)); 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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023