Real-rooted atlas

A110441 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(512\,n + 64\,n x + 32\,n x^{2} - 160\,n^{3} x^{2} - 40\,n^{3} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(1024 + 704\,x + 224\,x^{2} + 16\,x^{3} - 8\,x^{4} + 1024\,n + 1152\,n x + 192\,n x^{2} - 120\,n x^{3} - 44\,n x^{4} + 448\,n^{2} x - 352\,n^{2} x^{2} - 336\,n^{2} x^{3} - 72\,n^{2} x^{4} + n^{2} x^{5} - 320\,n^{3} x^{2} - 216\,n^{3} x^{3} - 36\,n^{3} x^{4} + 3\,n^{3} x^{5} - 16\,n^{4} x^{3} + 4\,n^{4} x^{4} + 3\,n^{4} x^{5} + 4\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(512 + 768\,x + 320\,x^{2} - 32\,x^{4} - 4\,x^{5} - 448\,n x^{2} - 608\,n x^{3} - 216\,n x^{4} - 20\,n x^{5} - 448\,n^{2} x^{2} - 608\,n^{2} x^{3} - 208\,n^{2} x^{4} - 12\,n^{2} x^{5} + n^{2} x^{6} + 16\,n^{3} x^{4} + 16\,n^{3} x^{5} + 2\,n^{3} x^{6} + 8\,n^{4} x^{4} + 8\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
21432955
34817610816348
4164832810256252087
55603632489624325003612056
6191215104261361792057508644967735
76528608001287361132805000011664137264372444
822288239104599472646144365500116928212662048812012831
97609692396826784003423744237160097070424147235840291610010724840
1025980835220481159358417164288141050007064064224459245465656862400012156469023
118870401327948848942144824204807849600046626048181104004678656797040860005324144294328764
12302854449622016202460256382361600414556000285410304131415648414597128995860132000012511469121691520762135
13103400961840432648236062721724534784209867200016453497608779169283281264648686180816214000208700817625687881967797937624
143530329667828940833036019207597604864102601616009032666112548834025623762534407447405681694528002771674431749122416701097622539717558791
Technical details
Node id
n-c10921741d0cb383 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-a71577c760e2f25d

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:     4,      4;
  2:    14,     32,       9;
  3:    48,    176,     108,      16;
  4:   164,    832,     810,     256,      25;
  5:   560,   3632,    4896,    2432,     500,     36;
  6:  1912,  15104,   26136,   17920,    5750,    864,     49;
  7:  6528,  60800,  128736,  113280,   50000,  11664,   1372,    64;
  8: 22288, 239104,  599472,  646144,  365500, 116928,  21266,  2048,   81;
  9: 76096, 923968, 2678400, 3423744, 2371600, 970704, 241472, 35840, 2916, 100;
%F Formula
The row polynomials satisfy (512*n+64*n*x-(160*n^3-32*n)*x^2-40*n^3*x^3+2*n^5*x^4)*P(n,x) = (1024*n+1024+(448*n^2+1152*n+704)*x-(320*n^3+352*n^2-192*n-224)*x^2-(16*n^4+216*n^3+336*n^2+120*n-16)*x^3+(4*n^5+4*n^4-36*n^3-72*n^2-44*n-8)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (512+768*x-(448*n^2+448*n-320)*x^2-(608*n^2+608*n)*x^3+(8*n^4+16*n^3-208*n^2-216*n-32)*x^4+(8*n^4+16*n^3-12*n^2-20*n-4)*x^5+(n^4+2*n^3+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[((1024*n + 1024 + (448*n^2 + 1152*n + 704)*x - (320*n^3 + 352*n^2 - 192*n - 224)*x^2 - (16*n^4 + 216*n^3 + 336*n^2 + 120*n - 16)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (512 + 768*x - (448*n^2 + 448*n - 320)*x^2 - (608*n^2 + 608*n)*x^3 + (8*n^4 + 16*n^3 - 208*n^2 - 216*n - 32)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(512*n + 64*n*x - (160*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*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] = ((1024*n + 1024 + (448*n^2 + 1152*n + 704)*x - (320*n^3 + 352*n^2 - 192*n - 224)*x^2 - (16*n^4 + 216*n^3 + 336*n^2 + 120*n - 16)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (512 + 768*x - (448*n^2 + 448*n - 320)*x^2 - (608*n^2 + 608*n)*x^3 + (8*n^4 + 16*n^3 - 208*n^2 - 216*n - 32)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(512*n + 64*n*x - (160*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*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