Real-rooted atlas

A008288 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 14\,x + 40\,x^{2} + 44\,x^{3} - 32\,x^{4} + 12\,n x - 92\,n x^{2} + 8\,n x^{3} + 128\,n x^{4} - 32\,n x^{5} + 44\,n^{2} x^{2} - 104\,n^{2} x^{3} - 64\,n^{2} x^{4} + 64\,n^{2} x^{5} + 48\,n^{3} x^{3} - 32\,n^{3} x^{4} - 32\,n^{3} x^{5} + 16\,n^{4} x^{4}\right)\,P_{n}(x) = \left(1 + 2\,x - 16\,x^{2} - 36\,x^{3} + 64\,x^{5} + 12\,n x + 44\,n x^{2} - 72\,n x^{3} - 256\,n x^{4} - 128\,n x^{5} + 44\,n^{2} x^{2} + 216\,n^{2} x^{3} + 96\,n^{2} x^{4} - 288\,n^{2} x^{5} - 128\,n^{2} x^{6} + 48\,n^{3} x^{3} + 224\,n^{3} x^{4} + 96\,n^{3} x^{5} - 128\,n^{3} x^{6} + 16\,n^{4} x^{4} + 64\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(2 - 12\,x - 32\,x^{2} - 8\,x^{3} + 64\,x^{4} + 24\,n x - 48\,n x^{2} - 320\,n x^{3} - 288\,n x^{4} + 128\,n x^{5} + 88\,n^{2} x^{2} + 112\,n^{2} x^{3} - 288\,n^{2} x^{4} - 352\,n^{2} x^{5} + 96\,n^{3} x^{3} + 192\,n^{3} x^{4} - 64\,n^{3} x^{5} - 128\,n^{3} x^{6} + 32\,n^{4} x^{4} + 64\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
221618
3580144229
412288115210242476
529928576011264640024381
670280023616727048960036864225654
71698112865443655687104006266882007041998185
8408228482949121587200424960058982404014080104857617115864

All 15 rows

Technical details
Node id
n-77a81144c7cca672 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-89ae647762cb208b

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:    2,     16;
   3:    5,     80,     144;
   4:   12,    288,    1152,     1024;
   5:   29,    928,    5760,    11264,     6400;
   6:   70,   2800,   23616,    72704,    89600,     36864;
   7:  169,   8112,   86544,   365568,   710400,    626688,    200704;
   8:  408,  22848,  294912,  1587200,  4249600,   5898240,   4014080,   1048576;
   9:  985,  63040,  955008,  6252544, 21414400,  40992768,  43753472,  24117248,   5308416;
  10: 2378, 171216, 2977920, 22980608, 96076800, 235708416, 346816512, 298844160, 138018816, 26214400;
%F Formula
The row polynomials satisfy (1+(12*n-14)*x+(44*n^2-92*n+40)*x^2+(48*n^3-104*n^2+8*n+44)*x^3+(16*n^4-32*n^3-64*n^2+128*n-32)*x^4-(32*n^3-64*n^2+32*n)*x^5)*P(n,x) = (1+(12*n+2)*x+(44*n^2+44*n-16)*x^2+(48*n^3+216*n^2-72*n-36)*x^3+(16*n^4+224*n^3+96*n^2-256*n)*x^4+(64*n^4+96*n^3-288*n^2-128*n+64)*x^5-(128*n^3+128*n^2)*x^6)*P(n-2,x) + (2+(24*n-12)*x+(88*n^2-48*n-32)*x^2+(96*n^3+112*n^2-320*n-8)*x^3+(32*n^4+192*n^3-288*n^2-288*n+64)*x^4+(64*n^4-64*n^3-352*n^2+128*n)*x^5-128*n^3*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 16*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (12*n + 2)*x + (44*n^2 + 44*n - 16)*x^2 + (48*n^3 + 216*n^2 - 72*n - 36)*x^3 + (16*n^4 + 224*n^3 + 96*n^2 - 256*n)*x^4 + (64*n^4 + 96*n^3 - 288*n^2 - 128*n + 64)*x^5 - (128*n^3 + 128*n^2)*x^6)*P[n - 2, x] + (2 + (24*n - 12)*x + (88*n^2 - 48*n - 32)*x^2 + (96*n^3 + 112*n^2 - 320*n - 8)*x^3 + (32*n^4 + 192*n^3 - 288*n^2 - 288*n + 64)*x^4 + (64*n^4 - 64*n^3 - 352*n^2 + 128*n)*x^5 - 128*n^3*x^6)*P[n - 1, x])/(1 + (12*n - 14)*x + (44*n^2 - 92*n + 40)*x^2 + (48*n^3 - 104*n^2 + 8*n + 44)*x^3 + (16*n^4 - 32*n^3 - 64*n^2 + 128*n - 32)*x^4 - (32*n^3 - 64*n^2 + 32*n)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 16*x; for(n = 3, N, my(i = n); v[i] = ((1 + (12*n + 2)*x + (44*n^2 + 44*n - 16)*x^2 + (48*n^3 + 216*n^2 - 72*n - 36)*x^3 + (16*n^4 + 224*n^3 + 96*n^2 - 256*n)*x^4 + (64*n^4 + 96*n^3 - 288*n^2 - 128*n + 64)*x^5 - (128*n^3 + 128*n^2)*x^6)*v[i - 2] + (2 + (24*n - 12)*x + (88*n^2 - 48*n - 32)*x^2 + (96*n^3 + 112*n^2 - 320*n - 8)*x^3 + (32*n^4 + 192*n^3 - 288*n^2 - 288*n + 64)*x^4 + (64*n^4 - 64*n^3 - 352*n^2 + 128*n)*x^5 - 128*n^3*x^6)*v[i - 1])/(1 + (12*n - 14)*x + (44*n^2 - 92*n + 40)*x^2 + (48*n^3 - 104*n^2 + 8*n + 44)*x^3 + (16*n^4 - 32*n^3 - 64*n^2 + 128*n - 32)*x^4 - (32*n^3 - 64*n^2 + 32*n)*x^5)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.