Real-rooted atlas

A081579 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 2\,x^{2} + 5\,n - 5\,n x - 2\,n x^{2} + 5\,n^{2} x\right)\,P_{n}(x) = \left(-70 - 170\,x - 72\,x^{2} - 8\,x^{3} + 105\,n + 225\,n x + 48\,n x^{2} - 25\,n^{2} - 45\,n^{2} x\right)\,P_{n-1}(x) + \left(-25 - 110\,x - 145\,x^{2} - 52\,x^{3} - 4\,x^{4} + 25\,n + 100\,n x + 105\,n x^{2} + 10\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
216824
37211224208
4320800480641664
5128044804800166416012384
64992224003456021504512038488960
7188161031682092801930248128014592896621056
869632451072113510413987848806402749443942420484251648
925344018944005698560882892875136003502080860160102400460828658176
1091136077209602703360050556928546099203459072012615680253952025804810240190846976
11324403230720000122757120269303808353955840286384128142069760421888007170048634880225281258450944
12114524161198817285384601601356546048210518016020848803841325678592534609920133152768195379201531904491528230961152
1340148992460357632229623398465356103681171058688013765705728107572756485567021056187755724840124416051724288363724810649653467209728
1413991936017440768009565962240303642705926174638080084186071040783961292804998103040021625012224623656960011642470401336934408519680229376345292111872
Technical details
Node id
n-e491b6b31274204e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c0cba542a6350c02

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      2;
  2:     16,       8;
  3:     72,     112,      24;
  4:    320,     800,     480,      64;
  5:   1280,    4480,    4800,    1664,     160;
  6:   4992,   22400,   34560,   21504,    5120,     384;
  7:  18816,  103168,  209280,  193024,   81280,   14592,    896;
  8:  69632,  451072, 1135104, 1398784,  880640,  274944,  39424,   2048;
  9: 253440, 1894400, 5698560, 8828928, 7513600, 3502080, 860160, 102400, 4608;
%F Formula
The row polynomials satisfy (5*n-5+(5*n^2-5*n)*x-(2*n-2)*x^2)*P(n,x) = -(25*n^2-105*n+70+(45*n^2-225*n+170)*x-(48*n-72)*x^2+8*x^3)*P(n-1,x) + (25*n-25+(100*n-110)*x+(105*n-145)*x^2+(10*n-52)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(25*n^2 - 105*n + 70 + (45*n^2 - 225*n + 170)*x - (48*n - 72)*x^2 + 8*x^3)*P[n - 1, x] + (25*n - 25 + (100*n - 110)*x + (105*n - 145)*x^2 + (10*n - 52)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(5*n - 5 + (5*n^2 - 5*n)*x - (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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(25*n^2 - 105*n + 70 + (45*n^2 - 225*n + 170)*x - (48*n - 72)*x^2 + 8*x^3)*v[i - 1] + (25*n - 25 + (100*n - 110)*x + (105*n - 145)*x^2 + (10*n - 52)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(5*n - 5 + (5*n^2 - 5*n)*x - (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

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.