Real-rooted atlas

A086645 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-18 + 78\,x - 18\,x^{2} + 33\,n - 199\,n x + 69\,n x^{2} - 20\,n^{2} + 188\,n^{2} x - 104\,n^{2} x^{2} + 4\,n^{3} - 78\,n^{3} x + 77\,n^{3} x^{2} + 12\,n^{4} x - 28\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-72 - 120\,x + 40\,x^{2} + 8\,x^{3} + 132\,n - 4\,n x - 496\,n x^{2} - 60\,n x^{3} - 80\,n^{2} + 272\,n^{2} x + 726\,n^{2} x^{2} + 16\,n^{3} - 216\,n^{3} x - 324\,n^{3} x^{2} + 126\,n^{3} x^{3} + 48\,n^{4} x + 8\,n^{4} x^{2} - 104\,n^{4} x^{3} + 16\,n^{5} x^{2} + 24\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-18\,x - 34\,x^{2} + 6\,x^{3} + 4\,x^{4} + 24\,n x - 8\,n x^{2} - 126\,n x^{3} - 28\,n x^{4} - 8\,n^{2} x + 56\,n^{2} x^{2} + 146\,n^{2} x^{3} + 4\,n^{2} x^{4} - 24\,n^{3} x^{2} - 32\,n^{3} x^{3} + 32\,n^{3} x^{4} - 8\,n^{4} x^{3} - 16\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
22810
386436108
4323846481281192
512820485760409640012432
6512102404032051200200001152123424
72048491522488324587523360008294431361180864
881922293761419264344064037632001806336307328819210982528
93276810485767667712230686723379200024772608842956810485762073699881216
1013107247185923981312014313062426357760026276659213908787235389440335923251200892025344
1152428820971520200540160838860800186368000023616552961722040320692060160136857600102400001239047847554048
122097152922746889861857284706009088122552320001889324236817679613952989646028831313018884956160002998476829491268168312832
138388608402653184475634073625501368320762052608001386133585921589704130561151588106245123948544013120307200170095411284934656692224585762967552
14335544321744830464225737441281343519457284532469760009510416547841294473363456115682259763267104748339224366284800051595608064558209433623397171216056324986412277760
1513421772875161927681056964608006914897346562599839334400618535688601697583376629761040455827456074894530314243568723558400107953887641619236755865617637867520629407744368640042101282750464
Technical details
Node id
n-f6a165d0ec5e0cba · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-beb8457930cf3c20

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:      2,       8;
   3:      8,      64,       36;
   4:     32,     384,      648,       128;
   5:    128,    2048,     5760,      4096,       400;
   6:    512,   10240,    40320,     51200,     20000,      1152;
   7:   2048,   49152,   248832,    458752,    336000,     82944,      3136;
   8:   8192,  229376,  1419264,   3440640,   3763200,   1806336,    307328,     8192;
   9:  32768, 1048576,  7667712,  23068672,  33792000,  24772608,   8429568,  1048576,   20736;
  10: 131072, 4718592, 39813120, 143130624, 263577600, 262766592, 139087872, 35389440, 3359232, 51200;
%F Formula
The row polynomials satisfy (4*n^3-20*n^2+33*n-18+(12*n^4-78*n^3+188*n^2-199*n+78)*x+(4*n^5-28*n^4+77*n^3-104*n^2+69*n-18)*x^2)*P(n,x) = (16*n^3-80*n^2+132*n-72+(48*n^4-216*n^3+272*n^2-4*n-120)*x+(16*n^5+8*n^4-324*n^3+726*n^2-496*n+40)*x^2+(24*n^5-104*n^4+126*n^3-60*n+8)*x^3)*P(n-1,x) - ((8*n^2-24*n+18)*x+(24*n^3-56*n^2+8*n+34)*x^2+(8*n^4+32*n^3-146*n^2+126*n-6)*x^3+(16*n^4-32*n^3-4*n^2+28*n-4)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 80*n^2 + 132*n - 72 + (48*n^4 - 216*n^3 + 272*n^2 - 4*n - 120)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (24*n^5 - 104*n^4 + 126*n^3 - 60*n + 8)*x^3)*P[n - 1, x] - ((8*n^2 - 24*n + 18)*x + (24*n^3 - 56*n^2 + 8*n + 34)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^3 - 20*n^2 + 33*n - 18 + (12*n^4 - 78*n^3 + 188*n^2 - 199*n + 78)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; for(n = 3, N, my(i = n); v[i] = ((16*n^3 - 80*n^2 + 132*n - 72 + (48*n^4 - 216*n^3 + 272*n^2 - 4*n - 120)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (24*n^5 - 104*n^4 + 126*n^3 - 60*n + 8)*x^3)*v[i - 1] - ((8*n^2 - 24*n + 18)*x + (24*n^3 - 56*n^2 + 8*n + 34)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^3 - 20*n^2 + 33*n - 18 + (12*n^4 - 78*n^3 + 188*n^2 - 199*n + 78)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*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