Real-rooted atlas

A008288 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,x + 40\,x^{2} + 28\,x^{3} - 16\,x^{4} - 10\,n + 22\,n x - 56\,n x^{2} - 44\,n x^{3} + 16\,n x^{4} + 4\,n^{2} - 24\,n^{2} x + 16\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{3} x\right)\,P_{n}(x) = \left(28 + 40\,x - 256\,x^{2} - 208\,x^{3} + 80\,x^{4} + 64\,x^{5} - 80\,n - 270\,n x + 52\,n x^{2} + 196\,n x^{3} + 16\,n x^{4} + 52\,n^{2} + 184\,n^{2} x + 32\,n^{2} x^{2} - 48\,n^{2} x^{3} - 8\,n^{3} - 24\,n^{3} x\right)\,P_{n-1}(x) + \left(3 + 14\,x - 24\,x^{2} - 132\,x^{3} + 20\,x^{4} + 160\,x^{5} + 64\,x^{6} - 8\,n - 56\,n x - 72\,n x^{2} + 144\,n x^{3} + 208\,n x^{4} + 64\,n x^{5} + 4\,n^{2} + 32\,n^{2} x + 72\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
325521693
48432832064796
52611512275217282566509
6770594816256188808704102451582
721972126078160136512115456419844096399665
8612071248330368779136989184655360196608163843044408

All 15 rows

Technical details
Node id
n-1607a06f087588c8 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e241b42e92753aa4

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     6,      4;
   3:    25,     52,      16;
   4:    84,    328,     320,       64;
   5:   261,   1512,    2752,     1728,      256;
   6:   770,   5948,   16256,    18880,     8704,     1024;
   7:  2197,  21260,   78160,   136512,   115456,    41984,     4096;
   8:  6120,  71248,  330368,   779136,   989184,   655360,   196608,    16384;
   9: 16745, 227984, 1278464,  3817088,  6550016,  6496256,  3530752,   901120,   65536;
  10: 45182, 704564, 4639744, 16808576, 36580352, 49014784, 39763968, 18300928, 4063232, 262144;
%F Formula
The row polynomials satisfy (4*n^2-10*n+6+(8*n^3-24*n^2+22*n-6)*x+(16*n^2-56*n+40)*x^2+(16*n^2-44*n+28)*x^3+(16*n-16)*x^4)*P(n,x) = -(8*n^3-52*n^2+80*n-28+(24*n^3-184*n^2+270*n-40)*x-(32*n^2+52*n-256)*x^2+(48*n^2-196*n+208)*x^3-(16*n+80)*x^4-64*x^5)*P(n-1,x) + (4*n^2-8*n+3+(32*n^2-56*n+14)*x+(72*n^2-72*n-24)*x^2+(32*n^2+144*n-132)*x^3+(208*n+20)*x^4+(64*n+160)*x^5+64*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 52*n^2 + 80*n - 28 + (24*n^3 - 184*n^2 + 270*n - 40)*x - (32*n^2 + 52*n - 256)*x^2 + (48*n^2 - 196*n + 208)*x^3 - (16*n + 80)*x^4 - 64*x^5)*P[n - 1, x] + (4*n^2 - 8*n + 3 + (32*n^2 - 56*n + 14)*x + (72*n^2 - 72*n - 24)*x^2 + (32*n^2 + 144*n - 132)*x^3 + (208*n + 20)*x^4 + (64*n + 160)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (8*n^3 - 24*n^2 + 22*n - 6)*x + (16*n^2 - 56*n + 40)*x^2 + (16*n^2 - 44*n + 28)*x^3 + (16*n - 16)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(8*n^3 - 52*n^2 + 80*n - 28 + (24*n^3 - 184*n^2 + 270*n - 40)*x - (32*n^2 + 52*n - 256)*x^2 + (48*n^2 - 196*n + 208)*x^3 - (16*n + 80)*x^4 - 64*x^5)*v[i - 1] + (4*n^2 - 8*n + 3 + (32*n^2 - 56*n + 14)*x + (72*n^2 - 72*n - 24)*x^2 + (32*n^2 + 144*n - 132)*x^3 + (208*n + 20)*x^4 + (64*n + 160)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (8*n^3 - 24*n^2 + 22*n - 6)*x + (16*n^2 - 56*n + 40)*x^2 + (16*n^2 - 44*n + 28)*x^3 + (16*n - 16)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.