Real-rooted atlas

A103327 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-n x - 2\,n^{2} + 14\,n^{2} x - 2\,n^{2} x^{2} + 4\,n^{3} - 36\,n^{3} x + 12\,n^{3} x^{2} + 24\,n^{4} x - 24\,n^{4} x^{2} + 16\,n^{5} x^{2}\right)\,P_{n}(x) = \left(24\,x^{2} - 24\,x^{3} - 4\,n x + 96\,n x^{2} + 112\,n x^{3} - 8\,n^{2} - 104\,n^{2} x - 304\,n^{2} x^{2} - 104\,n^{2} x^{3} + 16\,n^{3} + 48\,n^{3} x - 240\,n^{3} x^{2} - 304\,n^{3} x^{3} + 96\,n^{4} x + 384\,n^{4} x^{2} + 160\,n^{4} x^{3} + 64\,n^{5} x^{2} + 192\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x^{2} + 4\,x^{3} - 16\,x^{4} + 12\,n x^{2} + 64\,n x^{3} + 64\,n x^{4} - 8\,n^{2} x - 56\,n^{2} x^{2} - 104\,n^{2} x^{3} - 32\,n^{2} x^{4} - 48\,n^{3} x^{2} - 256\,n^{3} x^{3} - 256\,n^{3} x^{4} - 32\,n^{4} x^{3} - 128\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141620
216192144352
3641280345610245824
425671683456040960640089344
5102436864258048573440384000368641290240
64096180224165888055050247168000309657620070417813504
7163848519689732096432537608601600074317824224788481048576237715456

All 15 rows

Technical details
Node id
n-9e0c925faa90ea23 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6414dcf212314547

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,       16;
  2:     16,      192,       144;
  3:     64,     1280,      3456,       1024;
  4:    256,     7168,     34560,      40960,       6400;
  5:   1024,    36864,    258048,     573440,     384000,       36864;
  6:   4096,   180224,   1658880,    5505024,    7168000,     3096576,      200704;
  7:  16384,   851968,   9732096,   43253760,   86016000,    74317824,    22478848,    1048576;
  8:  65536,  3932160,  53673984,  299892736,  811008000,  1089994752,   674365440,  150994944,   5308416;
  9: 262144, 17825792, 283115520, 1908408320, 6560153600, 12145655808, 11868831744, 5536481280, 955514880, 26214400;
%F Formula
The row polynomials satisfy (4*n^3-2*n^2+(24*n^4-36*n^3+14*n^2-n)*x+(16*n^5-24*n^4+12*n^3-2*n^2)*x^2)*P(n,x) = (16*n^3-8*n^2+(96*n^4+48*n^3-104*n^2-4*n)*x+(64*n^5+384*n^4-240*n^3-304*n^2+96*n+24)*x^2+(192*n^5+160*n^4-304*n^3-104*n^2+112*n-24)*x^3)*P(n-1,x) - (8*n^2*x+(48*n^3+56*n^2-12*n-2)*x^2+(32*n^4+256*n^3+104*n^2-64*n-4)*x^3+(128*n^4+256*n^3+32*n^2-64*n+16)*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[((16*n^3 - 8*n^2 + (96*n^4 + 48*n^3 - 104*n^2 - 4*n)*x + (64*n^5 + 384*n^4 - 240*n^3 - 304*n^2 + 96*n + 24)*x^2 + (192*n^5 + 160*n^4 - 304*n^3 - 104*n^2 + 112*n - 24)*x^3)*P[n - 1, x] - (8*n^2*x + (48*n^3 + 56*n^2 - 12*n - 2)*x^2 + (32*n^4 + 256*n^3 + 104*n^2 - 64*n - 4)*x^3 + (128*n^4 + 256*n^3 + 32*n^2 - 64*n + 16)*x^4)*D[P[n - 1, x], x])/(4*n^3 - 2*n^2 + (24*n^4 - 36*n^3 + 14*n^2 - n)*x + (16*n^5 - 24*n^4 + 12*n^3 - 2*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] = ((16*n^3 - 8*n^2 + (96*n^4 + 48*n^3 - 104*n^2 - 4*n)*x + (64*n^5 + 384*n^4 - 240*n^3 - 304*n^2 + 96*n + 24)*x^2 + (192*n^5 + 160*n^4 - 304*n^3 - 104*n^2 + 112*n - 24)*x^3)*v[i - 1] - (8*n^2*x + (48*n^3 + 56*n^2 - 12*n - 2)*x^2 + (32*n^4 + 256*n^3 + 104*n^2 - 64*n - 4)*x^3 + (128*n^4 + 256*n^3 + 32*n^2 - 64*n + 16)*x^4)*deriv(v[i - 1], x))/(4*n^3 - 2*n^2 + (24*n^4 - 36*n^3 + 14*n^2 - n)*x + (16*n^5 - 24*n^4 + 12*n^3 - 2*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

Same fingerprint ?

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