Real-rooted atlas

A207824 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64\,n + 12\,n x + 9\,n x^{2} - 80\,n^{3} x^{2} - 30\,n^{3} x^{3} + 4\,n^{5} x^{4}\right)\,P_{n}(x) = \left(192 + 324\,x + 147\,x^{2} - 24\,x^{3} - 24\,x^{4} + 192\,n + 548\,n x + 123\,n x^{2} - 296\,n x^{3} - 132\,n x^{4} + 224\,n^{2} x - 264\,n^{2} x^{2} - 602\,n^{2} x^{3} - 216\,n^{2} x^{4} + 8\,n^{2} x^{5} - 240\,n^{3} x^{2} - 362\,n^{3} x^{3} - 108\,n^{3} x^{4} + 24\,n^{3} x^{5} - 32\,n^{4} x^{3} + 12\,n^{4} x^{4} + 24\,n^{4} x^{5} + 12\,n^{5} x^{4} + 8\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(64 + 144\,x + 76\,x^{2} - 42\,x^{3} - 50\,x^{4} - 12\,x^{5} - 224\,n x^{2} - 456\,n x^{3} - 292\,n x^{4} - 60\,n x^{5} - 224\,n^{2} x^{2} - 456\,n^{2} x^{3} - 276\,n^{2} x^{4} - 36\,n^{2} x^{5} + 8\,n^{2} x^{6} + 32\,n^{3} x^{4} + 48\,n^{3} x^{5} + 16\,n^{3} x^{6} + 16\,n^{4} x^{4} + 24\,n^{4} x^{5} + 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
161622
235192144371
32041696259210245516
41189132483067224576640076085
569309681630067236454419200036864997826
640391678528264470443008003424000132710420070412616231
723541646206082167430444247040472320002764800084295681048576155135512

All 15 rows

Technical details
Node id
n-31000305b71f8c7b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e36f9a64caaa62ff

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:       6,        16;
  2:      35,       192,        144;
  3:     204,      1696,       2592,       1024;
  4:    1189,     13248,      30672,      24576,       6400;
  5:    6930,     96816,     300672,     364544,     192000,      36864;
  6:   40391,    678528,    2644704,    4300800,    3424000,    1327104,     200704;
  7:  235416,   4620608,   21674304,   44247040,   47232000,   27648000,    8429568,    1048576;
  8: 1372105,  30811776,  168986592,  415309824,  556512000,  436617216,  200904704,   50331648,   5308416;
  9: 7997214, 202203728, 1269551232, 3649294336, 5887526400, 5797564416, 3574136832, 1350565888, 286654464, 26214400;
%F Formula
The row polynomials satisfy (64*n+12*n*x-(80*n^3-9*n)*x^2-30*n^3*x^3+4*n^5*x^4)*P(n,x) = (192*n+192+(224*n^2+548*n+324)*x-(240*n^3+264*n^2-123*n-147)*x^2-(32*n^4+362*n^3+602*n^2+296*n+24)*x^3+(12*n^5+12*n^4-108*n^3-216*n^2-132*n-24)*x^4+(8*n^5+24*n^4+24*n^3+8*n^2)*x^5)*P(n-1,x) + (64+144*x-(224*n^2+224*n-76)*x^2-(456*n^2+456*n+42)*x^3+(16*n^4+32*n^3-276*n^2-292*n-50)*x^4+(24*n^4+48*n^3-36*n^2-60*n-12)*x^5+(8*n^4+16*n^3+8*n^2)*x^6)*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[((192*n + 192 + (224*n^2 + 548*n + 324)*x - (240*n^3 + 264*n^2 - 123*n - 147)*x^2 - (32*n^4 + 362*n^3 + 602*n^2 + 296*n + 24)*x^3 + (12*n^5 + 12*n^4 - 108*n^3 - 216*n^2 - 132*n - 24)*x^4 + (8*n^5 + 24*n^4 + 24*n^3 + 8*n^2)*x^5)*P[n - 1, x] + (64 + 144*x - (224*n^2 + 224*n - 76)*x^2 - (456*n^2 + 456*n + 42)*x^3 + (16*n^4 + 32*n^3 - 276*n^2 - 292*n - 50)*x^4 + (24*n^4 + 48*n^3 - 36*n^2 - 60*n - 12)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*D[P[n - 1, x], x])/(64*n + 12*n*x - (80*n^3 - 9*n)*x^2 - 30*n^3*x^3 + 4*n^5*x^4)]];
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] = ((192*n + 192 + (224*n^2 + 548*n + 324)*x - (240*n^3 + 264*n^2 - 123*n - 147)*x^2 - (32*n^4 + 362*n^3 + 602*n^2 + 296*n + 24)*x^3 + (12*n^5 + 12*n^4 - 108*n^3 - 216*n^2 - 132*n - 24)*x^4 + (8*n^5 + 24*n^4 + 24*n^3 + 8*n^2)*x^5)*v[i - 1] + (64 + 144*x - (224*n^2 + 224*n - 76)*x^2 - (456*n^2 + 456*n + 42)*x^3 + (16*n^4 + 32*n^3 - 276*n^2 - 292*n - 50)*x^4 + (24*n^4 + 48*n^3 - 36*n^2 - 60*n - 12)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*deriv(v[i - 1], x))/(64*n + 12*n*x - (80*n^3 - 9*n)*x^2 - 30*n^3*x^3 + 4*n^5*x^4)); 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