Real-rooted atlas

A081578 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 - 32\,x - 24\,x^{2} - 10\,x^{3} + 2\,x^{4} + 192\,n x - 16\,n x^{2} - 32\,n x^{3} - 4\,n x^{4} + 176\,n^{2} x^{2} + 40\,n^{2} x^{3} - 6\,n^{2} x^{4} - n^{2} x^{5} + 48\,n^{3} x^{3} + 8\,n^{3} x^{4} - n^{3} x^{5} + 4\,n^{4} x^{4}\right)\,P_{n}(x) = \left(192 + 672\,x + 888\,x^{2} + 546\,x^{3} + 150\,x^{4} + 6\,x^{5} - 6\,x^{6} + 576\,n x + 1584\,n x^{2} + 1584\,n x^{3} + 648\,n x^{4} + 57\,n x^{5} - 15\,n x^{6} + 528\,n^{2} x^{2} + 1080\,n^{2} x^{3} + 678\,n^{2} x^{4} + 114\,n^{2} x^{5} - 12\,n^{2} x^{6} + 144\,n^{3} x^{3} + 216\,n^{3} x^{4} + 69\,n^{3} x^{5} - 3\,n^{3} x^{6} + 12\,n^{4} x^{4} + 12\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(128 + 192\,x + 48\,x^{2} - 100\,x^{3} - 48\,x^{4} - 3\,x^{5} - x^{6} + 384\,n x + 512\,n x^{2} + 240\,n x^{3} + 12\,n x^{4} - 8\,n x^{5} - 3\,n x^{6} + 352\,n^{2} x^{2} + 400\,n^{2} x^{3} + 140\,n^{2} x^{4} + 5\,n^{2} x^{5} - 3\,n^{2} x^{6} + 96\,n^{3} x^{3} + 80\,n^{3} x^{4} + 14\,n^{3} x^{5} - n^{3} x^{6} + 8\,n^{4} x^{4} + 4\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
2728944
32012010816264
461488702272251548
5182182036362416550368640
65476564167671585662259724946980
716402296071496881605020013392156864249480

All 15 rows

Technical details
Node id
n-bceaadda08814e49 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4659f198885a7703

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,      4;
  2:     7,     28,       9;
  3:    20,    120,     108,      16;
  4:    61,    488,     702,     272,      25;
  5:   182,   1820,    3636,    2416,     550,     36;
  6:   547,   6564,   16767,   15856,    6225,    972,     49;
  7:  1640,  22960,   71496,   88160,   50200,  13392,   1568,    64;
  8:  4921,  78736,  289260,  440416,  331150, 128880,  25480,  2368,   81;
  9: 14762, 265716, 1125144, 2040864, 1914300, 991656, 285768, 44352, 3402, 100;
%F Formula
The row polynomials satisfy (64+(192*n-32)*x+(176*n^2-16*n-24)*x^2+(48*n^3+40*n^2-32*n-10)*x^3+(4*n^4+8*n^3-6*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (192+(576*n+672)*x+(528*n^2+1584*n+888)*x^2+(144*n^3+1080*n^2+1584*n+546)*x^3+(12*n^4+216*n^3+678*n^2+648*n+150)*x^4+(12*n^4+69*n^3+114*n^2+57*n+6)*x^5-(3*n^3+12*n^2+15*n+6)*x^6)*P(n-2,x) + (128+(384*n+192)*x+(352*n^2+512*n+48)*x^2+(96*n^3+400*n^2+240*n-100)*x^3+(8*n^4+80*n^3+140*n^2+12*n-48)*x^4+(4*n^4+14*n^3+5*n^2-8*n-3)*x^5-(n^3+3*n^2+3*n+1)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((192 + (576*n + 672)*x + (528*n^2 + 1584*n + 888)*x^2 + (144*n^3 + 1080*n^2 + 1584*n + 546)*x^3 + (12*n^4 + 216*n^3 + 678*n^2 + 648*n + 150)*x^4 + (12*n^4 + 69*n^3 + 114*n^2 + 57*n + 6)*x^5 - (3*n^3 + 12*n^2 + 15*n + 6)*x^6)*P[n - 2, x] + (128 + (384*n + 192)*x + (352*n^2 + 512*n + 48)*x^2 + (96*n^3 + 400*n^2 + 240*n - 100)*x^3 + (8*n^4 + 80*n^3 + 140*n^2 + 12*n - 48)*x^4 + (4*n^4 + 14*n^3 + 5*n^2 - 8*n - 3)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(64 + (192*n - 32)*x + (176*n^2 - 16*n - 24)*x^2 + (48*n^3 + 40*n^2 - 32*n - 10)*x^3 + (4*n^4 + 8*n^3 - 6*n^2 - 4*n + 2)*x^4 - (n^3 + 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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((192 + (576*n + 672)*x + (528*n^2 + 1584*n + 888)*x^2 + (144*n^3 + 1080*n^2 + 1584*n + 546)*x^3 + (12*n^4 + 216*n^3 + 678*n^2 + 648*n + 150)*x^4 + (12*n^4 + 69*n^3 + 114*n^2 + 57*n + 6)*x^5 - (3*n^3 + 12*n^2 + 15*n + 6)*x^6)*v[i - 2] + (128 + (384*n + 192)*x + (352*n^2 + 512*n + 48)*x^2 + (96*n^3 + 400*n^2 + 240*n - 100)*x^3 + (8*n^4 + 80*n^3 + 140*n^2 + 12*n - 48)*x^4 + (4*n^4 + 14*n^3 + 5*n^2 - 8*n - 3)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(64 + (192*n - 32)*x + (176*n^2 - 16*n - 24)*x^2 + (48*n^3 + 40*n^2 - 32*n - 10)*x^3 + (4*n^4 + 8*n^3 - 6*n^2 - 4*n + 2)*x^4 - (n^3 + 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

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.