Real-rooted atlas

A207607 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(34 + 120\,x + 168\,x^{2} + 64\,x^{3} + 52\,n + 32\,n x - 144\,n x^{2} - 64\,n x^{3} - 120\,n^{2} - 192\,n^{2} x + 16\,n^{3} + 64\,n^{3} x\right)\,P_{n}(x) = \left(188 + 724\,x + 848\,x^{2} + 400\,x^{3} + 64\,x^{4} - 170\,n - 484\,n x - 584\,n x^{2} - 304\,n x^{3} - 64\,n x^{4} - 168\,n^{2} - 256\,n^{2} x - 64\,n^{2} x^{2} + 24\,n^{3} + 112\,n^{3} x + 64\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-205 - 972\,x - 1656\,x^{2} - 1280\,x^{3} - 464\,x^{4} - 64\,x^{5} - 80\,n - 272\,n x - 256\,n x^{2} - 64\,n x^{3} + 20\,n^{2} + 128\,n^{2} x + 208\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
22634868
38718610816397
4277830808296322243
58543302458428247526412380
625731220422140195368736182412867141
77619428489625211148871440249924288256359183

All 15 rows

Technical details
Node id
n-865a387bce2743c0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-aa322d6ba5da47fd

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:     7,      4;
  2:    26,     34,       8;
  3:    87,    186,     108,      16;
  4:   277,    830,     808,     296,      32;
  5:   854,   3302,    4584,    2824,     752,      64;
  6:  2573,  12204,   22140,   19536,    8736,    1824,    128;
  7:  7619,  42848,   96252,  111488,   71440,   24992,   4288,    256;
  8: 22258, 144842,  388528,  559528,  471280,  236064,  67712,   9856,   512;
  9: 64323, 475542, 1484184, 2561448, 2686896, 1769184, 725760, 176256, 22272, 1024;
%F Formula
The row polynomials satisfy (16*n^3-120*n^2+52*n+34+(64*n^3-192*n^2+32*n+120)*x-(144*n-168)*x^2-(64*n-64)*x^3)*P(n,x) = (24*n^3-168*n^2-170*n+188+(112*n^3-256*n^2-484*n+724)*x+(64*n^3-64*n^2-584*n+848)*x^2-(304*n-400)*x^3-(64*n-64)*x^4)*P(n-1,x) + (20*n^2-80*n-205+(128*n^2-272*n-972)*x+(208*n^2-256*n-1656)*x^2+(64*n^2-64*n-1280)*x^3-464*x^4-64*x^5)*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[((24*n^3 - 168*n^2 - 170*n + 188 + (112*n^3 - 256*n^2 - 484*n + 724)*x + (64*n^3 - 64*n^2 - 584*n + 848)*x^2 - (304*n - 400)*x^3 - (64*n - 64)*x^4)*P[n - 1, x] + (20*n^2 - 80*n - 205 + (128*n^2 - 272*n - 972)*x + (208*n^2 - 256*n - 1656)*x^2 + (64*n^2 - 64*n - 1280)*x^3 - 464*x^4 - 64*x^5)*D[P[n - 1, x], x])/(16*n^3 - 120*n^2 + 52*n + 34 + (64*n^3 - 192*n^2 + 32*n + 120)*x - (144*n - 168)*x^2 - (64*n - 64)*x^3)]];
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] = ((24*n^3 - 168*n^2 - 170*n + 188 + (112*n^3 - 256*n^2 - 484*n + 724)*x + (64*n^3 - 64*n^2 - 584*n + 848)*x^2 - (304*n - 400)*x^3 - (64*n - 64)*x^4)*v[i - 1] + (20*n^2 - 80*n - 205 + (128*n^2 - 272*n - 972)*x + (208*n^2 - 256*n - 1656)*x^2 + (64*n^2 - 64*n - 1280)*x^3 - 464*x^4 - 64*x^5)*deriv(v[i - 1], x))/(16*n^3 - 120*n^2 + 52*n + 34 + (64*n^3 - 192*n^2 + 32*n + 120)*x - (144*n - 168)*x^2 - (64*n - 64)*x^3)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023