Real-rooted atlas

A085841 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 4\,x + 2\,n - 44\,n x + 112\,n^{2} x - 16\,n^{2} x^{2} - 80\,n^{3} x + 96\,n^{3} x^{2} - 192\,n^{4} x^{2} + 128\,n^{5} x^{2}\right)\,P_{n}(x) = \left(1 - 84\,x - 592\,x^{2} - 192\,x^{3} + 2\,n + 124\,n x + 1424\,n x^{2} + 896\,n x^{3} + 32\,n^{2} x + 784\,n^{2} x^{2} - 832\,n^{2} x^{3} - 80\,n^{3} x - 1760\,n^{3} x^{2} - 2432\,n^{3} x^{3} - 64\,n^{4} x^{2} + 1280\,n^{4} x^{3} + 128\,n^{5} x^{2} + 1536\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 40\,x^{2} - 192\,x^{3} - 128\,x^{4} - 96\,n x^{2} + 256\,n x^{3} + 512\,n x^{4} - 160\,n^{2} x^{2} + 704\,n^{2} x^{3} - 256\,n^{2} x^{4} + 512\,n^{3} x^{3} - 2048\,n^{3} x^{4} + 256\,n^{4} x^{3} - 1024\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131619
25160144309
37560302410244615
4913441814436864640062761
51126406652833792035200036864795963
61345761853281757184457600028753922007049599197
715728043243265894403203200050319360210739201048576111503023
817108808910721991475215558400045622886447767552014260633653084161258219857
919155041674432515973125912192002786254848544550092840642805769077391362621440013874496355
10212128029302561190707201881152000130025226244084125696056899928064317708697605505024000126877696150149653381
1123283364845456251040768523001600049843003392229618622464514131492864535868669952232128512000321000570886039797761599780268119
12253680076507204922368001307504000016431759360010437210112003427543285760574145003520046425702400001605002854400181193932800283534950416822209266809
13274680011625120909342720299956800004806289612804025781043200182282983833604478331027456058197934080000375570667929601059984506880099520767590413153337344174912142312075
1429584641710072015982387206409603200012753944985601362055919616081326254325760275485211688960525074694144000544577468497920286902473195520673423860695045340254961664603979776001801070805985581
Technical details
Node id
n-584d62f138731882 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c37f57020308301f

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:  3,    16;
  2:  5,   160,     144;
  3:  7,   560,    3024,     1024;
  4:  9,  1344,   18144,    36864,      6400;
  5: 11,  2640,   66528,   337920,    352000,      36864;
  6: 13,  4576,  185328,  1757184,   4576000,    2875392,     200704;
  7: 15,  7280,  432432,  6589440,  32032000,   50319360,   21073920,    1048576;
  8: 17, 10880,  891072, 19914752, 155584000,  456228864,  477675520,  142606336,   5308416;
  9: 19, 15504, 1674432, 51597312, 591219200, 2786254848, 5445500928, 4064280576, 907739136, 26214400;
%F Formula
The row polynomials satisfy (2*n-1-(80*n^3-112*n^2+44*n-4)*x+(128*n^5-192*n^4+96*n^3-16*n^2)*x^2)*P(n,x) = (2*n+1-(80*n^3-32*n^2-124*n+84)*x+(128*n^5-64*n^4-1760*n^3+784*n^2+1424*n-592)*x^2+(1536*n^5+1280*n^4-2432*n^3-832*n^2+896*n-192)*x^3)*P(n-1,x) + (4*x-(160*n^2+96*n-40)*x^2+(256*n^4+512*n^3+704*n^2+256*n-192)*x^3-(1024*n^4+2048*n^3+256*n^2-512*n+128)*x^4)*d/dx 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[((2*n + 1 - (80*n^3 - 32*n^2 - 124*n + 84)*x + (128*n^5 - 64*n^4 - 1760*n^3 + 784*n^2 + 1424*n - 592)*x^2 + (1536*n^5 + 1280*n^4 - 2432*n^3 - 832*n^2 + 896*n - 192)*x^3)*P[n - 1, x] + (4*x - (160*n^2 + 96*n - 40)*x^2 + (256*n^4 + 512*n^3 + 704*n^2 + 256*n - 192)*x^3 - (1024*n^4 + 2048*n^3 + 256*n^2 - 512*n + 128)*x^4)*D[P[n - 1, x], x])/(2*n - 1 - (80*n^3 - 112*n^2 + 44*n - 4)*x + (128*n^5 - 192*n^4 + 96*n^3 - 16*n^2)*x^2)]];
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] = ((2*n + 1 - (80*n^3 - 32*n^2 - 124*n + 84)*x + (128*n^5 - 64*n^4 - 1760*n^3 + 784*n^2 + 1424*n - 592)*x^2 + (1536*n^5 + 1280*n^4 - 2432*n^3 - 832*n^2 + 896*n - 192)*x^3)*v[i - 1] + (4*x - (160*n^2 + 96*n - 40)*x^2 + (256*n^4 + 512*n^3 + 704*n^2 + 256*n - 192)*x^3 - (1024*n^4 + 2048*n^3 + 256*n^2 - 512*n + 128)*x^4)*deriv(v[i - 1], x))/(2*n - 1 - (80*n^3 - 112*n^2 + 44*n - 4)*x + (128*n^5 - 192*n^4 + 96*n^3 - 16*n^2)*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

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