Real-rooted atlas

delannoy → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x - 8\,x^{2} - 4\,x^{3} + 16\,x^{4} + 12\,n x - 4\,n x^{2} - 56\,n x^{3} - 32\,n x^{4} + 44\,n^{2} x^{2} + 40\,n^{2} x^{3} - 64\,n^{2} x^{4} - 32\,n^{2} x^{5} + 48\,n^{3} x^{3} + 32\,n^{3} x^{4} - 32\,n^{3} x^{5} + 16\,n^{4} x^{4}\right)\,P_{n}(x) = \left(1 + 14\,x + 72\,x^{2} + 156\,x^{3} + 80\,x^{4} - 192\,x^{5} - 256\,x^{6} + 12\,n x + 132\,n x^{2} + 504\,n x^{3} + 672\,n x^{4} - 160\,n x^{5} - 640\,n x^{6} + 44\,n^{2} x^{2} + 360\,n^{2} x^{3} + 864\,n^{2} x^{4} + 384\,n^{2} x^{5} - 512\,n^{2} x^{6} + 48\,n^{3} x^{3} + 288\,n^{3} x^{4} + 352\,n^{3} x^{5} - 128\,n^{3} x^{6} + 16\,n^{4} x^{4} + 64\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(2 + 12\,x + 8\,x^{2} - 120\,x^{3} - 288\,x^{4} - 224\,x^{5} - 128\,x^{6} + 24\,n x + 128\,n x^{2} + 192\,n x^{3} - 160\,n x^{4} - 512\,n x^{5} - 384\,n x^{6} + 88\,n^{2} x^{2} + 400\,n^{2} x^{3} + 480\,n^{2} x^{4} - 160\,n^{2} x^{5} - 384\,n^{2} x^{6} + 96\,n^{3} x^{3} + 320\,n^{3} x^{4} + 192\,n^{3} x^{5} - 128\,n^{3} x^{6} + 32\,n^{4} x^{4} + 64\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121618
2580144229
312288115210242476
429928576011264640024381
570280023616727048960036864225654
61698112865443655687104006266882007041998185
7408228482949121587200424960058982404014080104857617115864

All 15 rows

Technical details
Node id
n-30895e5d03a9a6e2 · 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 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:    2,     16;
  2:    5,     80,     144;
  3:   12,    288,    1152,     1024;
  4:   29,    928,    5760,    11264,     6400;
  5:   70,   2800,   23616,    72704,    89600,     36864;
  6:  169,   8112,   86544,   365568,   710400,    626688,    200704;
  7:  408,  22848,  294912,  1587200,  4249600,   5898240,   4014080,   1048576;
  8:  985,  63040,  955008,  6252544, 21414400,  40992768,  43753472,  24117248,   5308416;
  9: 2378, 171216, 2977920, 22980608, 96076800, 235708416, 346816512, 298844160, 138018816, 26214400;
%F Formula
The row polynomials satisfy (1+(12*n-2)*x+(44*n^2-4*n-8)*x^2+(48*n^3+40*n^2-56*n-4)*x^3+(16*n^4+32*n^3-64*n^2-32*n+16)*x^4-(32*n^3+32*n^2)*x^5)*P(n,x) = (1+(12*n+14)*x+(44*n^2+132*n+72)*x^2+(48*n^3+360*n^2+504*n+156)*x^3+(16*n^4+288*n^3+864*n^2+672*n+80)*x^4+(64*n^4+352*n^3+384*n^2-160*n-192)*x^5-(128*n^3+512*n^2+640*n+256)*x^6)*P(n-2,x) + (2+(24*n+12)*x+(88*n^2+128*n+8)*x^2+(96*n^3+400*n^2+192*n-120)*x^3+(32*n^4+320*n^3+480*n^2-160*n-288)*x^4+(64*n^4+192*n^3-160*n^2-512*n-224)*x^5-(128*n^3+384*n^2+384*n+128)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (12*n + 14)*x + (44*n^2 + 132*n + 72)*x^2 + (48*n^3 + 360*n^2 + 504*n + 156)*x^3 + (16*n^4 + 288*n^3 + 864*n^2 + 672*n + 80)*x^4 + (64*n^4 + 352*n^3 + 384*n^2 - 160*n - 192)*x^5 - (128*n^3 + 512*n^2 + 640*n + 256)*x^6)*P[n - 2, x] + (2 + (24*n + 12)*x + (88*n^2 + 128*n + 8)*x^2 + (96*n^3 + 400*n^2 + 192*n - 120)*x^3 + (32*n^4 + 320*n^3 + 480*n^2 - 160*n - 288)*x^4 + (64*n^4 + 192*n^3 - 160*n^2 - 512*n - 224)*x^5 - (128*n^3 + 384*n^2 + 384*n + 128)*x^6)*P[n - 1, x])/(1 + (12*n - 2)*x + (44*n^2 - 4*n - 8)*x^2 + (48*n^3 + 40*n^2 - 56*n - 4)*x^3 + (16*n^4 + 32*n^3 - 64*n^2 - 32*n + 16)*x^4 - (32*n^3 + 32*n^2)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = ((1 + (12*n + 14)*x + (44*n^2 + 132*n + 72)*x^2 + (48*n^3 + 360*n^2 + 504*n + 156)*x^3 + (16*n^4 + 288*n^3 + 864*n^2 + 672*n + 80)*x^4 + (64*n^4 + 352*n^3 + 384*n^2 - 160*n - 192)*x^5 - (128*n^3 + 512*n^2 + 640*n + 256)*x^6)*v[i - 2] + (2 + (24*n + 12)*x + (88*n^2 + 128*n + 8)*x^2 + (96*n^3 + 400*n^2 + 192*n - 120)*x^3 + (32*n^4 + 320*n^3 + 480*n^2 - 160*n - 288)*x^4 + (64*n^4 + 192*n^3 - 160*n^2 - 512*n - 224)*x^5 - (128*n^3 + 384*n^2 + 384*n + 128)*x^6)*v[i - 1])/(1 + (12*n - 2)*x + (44*n^2 - 4*n - 8)*x^2 + (48*n^3 + 40*n^2 - 56*n - 4)*x^3 + (16*n^4 + 32*n^3 - 64*n^2 - 32*n + 16)*x^4 - (32*n^3 + 32*n^2)*x^5)); 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

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.