Real-rooted atlas

A035607 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(5 + 42\,x + 94\,x^{2} + 16\,x^{3} - 32\,x^{4} - 2\,n - 18\,n x - 112\,n x^{2} - 44\,n x^{3} + 16\,n x^{4} - 16\,n^{2} x + 32\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{3} x\right)\,P_{n}(x) = \left(-5 - 42\,x - 94\,x^{2} - 16\,x^{3} + 32\,x^{4} - 8\,n + 10\,n x + 156\,n x^{2} + 52\,n x^{3} - 48\,n x^{4} + 24\,n^{2} + 72\,n^{2} x - 48\,n^{2} x^{2} - 48\,n^{2} x^{3} - 8\,n^{3} - 24\,n^{3} x\right)\,P_{n-1}(x) + \left(-1 - 10\,x - 24\,x^{2} + 44\,x^{3} + 228\,x^{4} + 224\,x^{5} + 64\,x^{6} + 16\,n x + 136\,n x^{2} + 352\,n x^{3} + 272\,n x^{4} + 64\,n x^{5} + 4\,n^{2} + 32\,n^{2} x + 72\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
211819
3399632167
41295606081281425
539324644896332851211593
611479432276163443216896204891571
7324733088128704238592214016819208192707759

All 15 rows

Technical details
Node id
n-d37a098e77d87c25 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3082d4b4b19b6c51

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:     1;
  2:    11,      8;
  3:    39,     96,      32;
  4:   129,    560,     608,     128;
  5:   393,   2464,    4896,    3328,      512;
  6:  1147,   9432,   27616,   34432,    16896,     2048;
  7:  3247,  33088,  128704,  238592,   214016,    81920,    8192;
  8:  8993, 109408,  532032, 1319680,  1764352,  1228800,  385024,   32768;
  9: 24497, 346560, 2024896, 6314496, 11335680, 11763712, 6676480, 1769472, 131072;
%F Formula
The row polynomials satisfy -(2*n-5-(8*n^3-16*n^2-18*n+42)*x-(32*n^2-112*n+94)*x^2-(16*n^2-44*n+16)*x^3-(16*n-32)*x^4)*P(n,x) = -(8*n^3-24*n^2+8*n+5+(24*n^3-72*n^2-10*n+42)*x+(48*n^2-156*n+94)*x^2+(48*n^2-52*n+16)*x^3+(48*n-32)*x^4)*P(n-1,x) + (4*n^2-1+(32*n^2+16*n-10)*x+(72*n^2+136*n-24)*x^2+(32*n^2+352*n+44)*x^3+(272*n+228)*x^4+(64*n+224)*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[(-(8*n^3 - 24*n^2 + 8*n + 5 + (24*n^3 - 72*n^2 - 10*n + 42)*x + (48*n^2 - 156*n + 94)*x^2 + (48*n^2 - 52*n + 16)*x^3 + (48*n - 32)*x^4)*P[n - 1, x] + (4*n^2 - 1 + (32*n^2 + 16*n - 10)*x + (72*n^2 + 136*n - 24)*x^2 + (32*n^2 + 352*n + 44)*x^3 + (272*n + 228)*x^4 + (64*n + 224)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(-2*n + 5 + (8*n^3 - 16*n^2 - 18*n + 42)*x + (32*n^2 - 112*n + 94)*x^2 + (16*n^2 - 44*n + 16)*x^3 + (16*n - 32)*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] = (-(8*n^3 - 24*n^2 + 8*n + 5 + (24*n^3 - 72*n^2 - 10*n + 42)*x + (48*n^2 - 156*n + 94)*x^2 + (48*n^2 - 52*n + 16)*x^3 + (48*n - 32)*x^4)*v[i - 1] + (4*n^2 - 1 + (32*n^2 + 16*n - 10)*x + (72*n^2 + 136*n - 24)*x^2 + (32*n^2 + 352*n + 44)*x^3 + (272*n + 228)*x^4 + (64*n + 224)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(-2*n + 5 + (8*n^3 - 16*n^2 - 18*n + 42)*x + (32*n^2 - 112*n + 94)*x^2 + (16*n^2 - 44*n + 16)*x^3 + (16*n - 32)*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.