Real-rooted atlas

A059473 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6\,n x^{2} + 2\,n x^{3} + 4\,n x^{4} + 3\,n^{2} + 3\,n^{2} x + 8\,n^{2} x^{2} + 4\,n^{2} x^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(-12 - 48\,x - 38\,x^{2} + 44\,x^{3} + 56\,x^{4} + 16\,x^{5} + 21\,n + 78\,n x + 86\,n x^{2} + 28\,n x^{3} + 24\,n^{2} + 42\,n^{2} x - 8\,n^{2} x^{2} - 16\,n^{2} x^{3} - 9\,n^{3} - 12\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x + 78\,x^{2} + 188\,x^{3} + 204\,x^{4} + 96\,x^{5} + 16\,x^{6} + 9\,n + 60\,n x + 150\,n x^{2} + 172\,n x^{3} + 88\,n x^{4} + 16\,n x^{5} + 9\,n^{2} + 36\,n^{2} x + 42\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2546816138
3320688416641488
4178054885680224025615444
59504383845728038592112641024156048
6493362470244850244716162344965427240961545864
7250880150105636666884699136336998413250562539521638415083136

All 15 rows

Technical details
Node id
n-a6fe927f63c6624f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-07711d6335840451

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:       8,        4;
  2:      54,       68,        16;
  3:     320,      688,       416,        64;
  4:    1780,     5488,      5680,      2240,       256;
  5:    9504,    38384,     57280,     38592,     11264,      1024;
  6:   49336,   247024,    485024,    471616,    234496,     54272,      4096;
  7:  250880,  1501056,   3666688,   4699136,   3369984,   1325056,    253952,    16384;
  8: 1255824,  8746368,  25601664,  40891392,  38666496,  21903360,   7114752,  1163264,   65536;
  9: 6208640, 49357632, 168552960, 323120640, 381087744, 284574720, 133005312, 36782080, 5242880, 262144;
%F Formula
The row polynomials satisfy (3*n^2+(3*n^3+3*n^2)*x+(8*n^2-6*n)*x^2+(4*n^2+2*n)*x^3+4*n*x^4)*P(n,x) = -(9*n^3-24*n^2-21*n+12+(12*n^3-42*n^2-78*n+48)*x+(8*n^2-86*n+38)*x^2+(16*n^2-28*n-44)*x^3-56*x^4-16*x^5)*P(n-1,x) + (9*n^2+9*n+(36*n^2+60*n+12)*x+(42*n^2+150*n+78)*x^2+(12*n^2+172*n+188)*x^3+(88*n+204)*x^4+(16*n+96)*x^5+16*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[(-(9*n^3 - 24*n^2 - 21*n + 12 + (12*n^3 - 42*n^2 - 78*n + 48)*x + (8*n^2 - 86*n + 38)*x^2 + (16*n^2 - 28*n - 44)*x^3 - 56*x^4 - 16*x^5)*P[n - 1, x] + (9*n^2 + 9*n + (36*n^2 + 60*n + 12)*x + (42*n^2 + 150*n + 78)*x^2 + (12*n^2 + 172*n + 188)*x^3 + (88*n + 204)*x^4 + (16*n + 96)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(3*n^2 + (3*n^3 + 3*n^2)*x + (8*n^2 - 6*n)*x^2 + (4*n^2 + 2*n)*x^3 + 4*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] = (-(9*n^3 - 24*n^2 - 21*n + 12 + (12*n^3 - 42*n^2 - 78*n + 48)*x + (8*n^2 - 86*n + 38)*x^2 + (16*n^2 - 28*n - 44)*x^3 - 56*x^4 - 16*x^5)*v[i - 1] + (9*n^2 + 9*n + (36*n^2 + 60*n + 12)*x + (42*n^2 + 150*n + 78)*x^2 + (12*n^2 + 172*n + 188)*x^3 + (88*n + 204)*x^4 + (16*n + 96)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(3*n^2 + (3*n^3 + 3*n^2)*x + (8*n^2 - 6*n)*x^2 + (4*n^2 + 2*n)*x^3 + 4*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.