Real-rooted atlas

A081580 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6\,n - 6\,n x - 88\,n x^{2} + 52\,n x^{3} + 16\,n x^{4} + 12\,n^{2} + 80\,n^{2} x^{2} + 16\,n^{2} x^{3} + 24\,n^{3} x\right)\,P_{n}(x) = \left(-84 - 642\,x - 1076\,x^{2} + 708\,x^{3} + 480\,x^{4} + 64\,x^{5} - 36\,n + 318\,n x + 1740\,n x^{2} - 156\,n x^{3} - 112\,n x^{4} + 228\,n^{2} + 720\,n^{2} x - 640\,n^{2} x^{2} - 176\,n^{2} x^{3} - 72\,n^{3} - 264\,n^{3} x\right)\,P_{n-1}(x) + \left(-9 - 78\,x - 72\,x^{2} + 908\,x^{3} + 2276\,x^{4} + 736\,x^{5} + 64\,x^{6} + 120\,n x + 984\,n x^{2} + 2192\,n x^{3} + 720\,n x^{4} + 64\,n x^{5} + 36\,n^{2} + 288\,n^{2} x + 600\,n^{2} x^{2} + 96\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2456816129
3196680448641388
490947766400249625614837
53762271006080047040128001024152526
6154571411484445605724803008006246440961541005
76132068033627902725208960449536017612802949121638415308824

All 15 rows

Technical details
Node id
n-efb58d16dbf8972f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4ef99950ffdb4802

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,        4;
  2:     45,       68,       16;
  3:    196,      680,      448,        64;
  4:    909,     4776,     6400,      2496,       256;
  5:   3762,    27100,    60800,     47040,     12800,      1024;
  6:  15457,   141148,   444560,    572480,    300800,     62464,      4096;
  7:  61320,   680336,  2790272,   5208960,   4495360,   1761280,    294912,    16384;
  8: 240057,  3135120, 15862656,  39178368,  49390080,  31309824,   9707520,  1359872,   65536;
  9: 924958, 13906932, 83873280, 259393152, 438438912, 405129216, 200392704, 51200000, 6160384, 262144;
%F Formula
The row polynomials satisfy (12*n^2-6*n+(24*n^3-6*n)*x+(80*n^2-88*n)*x^2+(16*n^2+52*n)*x^3+16*n*x^4)*P(n,x) = -(72*n^3-228*n^2+36*n+84+(264*n^3-720*n^2-318*n+642)*x+(640*n^2-1740*n+1076)*x^2+(176*n^2+156*n-708)*x^3+(112*n-480)*x^4-64*x^5)*P(n-1,x) + (36*n^2-9+(288*n^2+120*n-78)*x+(600*n^2+984*n-72)*x^2+(96*n^2+2192*n+908)*x^3+(720*n+2276)*x^4+(64*n+736)*x^5+64*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[(-(72*n^3 - 228*n^2 + 36*n + 84 + (264*n^3 - 720*n^2 - 318*n + 642)*x + (640*n^2 - 1740*n + 1076)*x^2 + (176*n^2 + 156*n - 708)*x^3 + (112*n - 480)*x^4 - 64*x^5)*P[n - 1, x] + (36*n^2 - 9 + (288*n^2 + 120*n - 78)*x + (600*n^2 + 984*n - 72)*x^2 + (96*n^2 + 2192*n + 908)*x^3 + (720*n + 2276)*x^4 + (64*n + 736)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(12*n^2 - 6*n + (24*n^3 - 6*n)*x + (80*n^2 - 88*n)*x^2 + (16*n^2 + 52*n)*x^3 + 16*n*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] = (-(72*n^3 - 228*n^2 + 36*n + 84 + (264*n^3 - 720*n^2 - 318*n + 642)*x + (640*n^2 - 1740*n + 1076)*x^2 + (176*n^2 + 156*n - 708)*x^3 + (112*n - 480)*x^4 - 64*x^5)*v[i - 1] + (36*n^2 - 9 + (288*n^2 + 120*n - 78)*x + (600*n^2 + 984*n - 72)*x^2 + (96*n^2 + 2192*n + 908)*x^3 + (720*n + 2276)*x^4 + (64*n + 736)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(12*n^2 - 6*n + (24*n^3 - 6*n)*x + (80*n^2 - 88*n)*x^2 + (16*n^2 + 52*n)*x^3 + 16*n*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. [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.