Real-rooted atlas

A060187 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 12\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(3 - 20\,x + 585\,x^{2} - 8\,n - 152\,n x - 1824\,n x^{2} + 4\,n^{2} + 144\,n^{2} x + 1976\,n^{2} x^{2} - 32\,n^{3} x - 896\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-52\,x^{2} + 612\,x^{3} + 32\,n x + 8\,n x^{2} - 1752\,n x^{3} - 16\,n^{2} x - 48\,n^{2} x^{2} + 1664\,n^{2} x^{3} + 32\,n^{3} x^{2} - 608\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-4\,x^{2} - 104\,x^{3} + 108\,x^{4} + 32\,n x^{2} + 256\,n x^{3} - 288\,n x^{4} - 16\,n^{2} x^{2} - 224\,n^{2} x^{3} + 240\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 - 108\,x - 4\,n + 216\,n x - 144\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(20\,x + 36\,x^{2} - 40\,n x - 120\,n x^{2} + 16\,n^{2} x + 112\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2145
3124934
419220716316
5130420701216253616
61948151382691259253648960
712888948873767682635752599249764160
8187165459134155568649307521836521067716413493760
9126208298450839764608116875350894703681624898841932881265789440
101786921580706034732755217353323502498878584106369062811240576015935131005775114240
111236152820452692852814464226730820505497532884844439240818114112578567384074215903800121137178316800
12170854042032184322449291600270920577450102916871330414008129708866935566782721136495387254670242700214333351443535670476800
13121257122135087730171342579008303459450517517172740319264364211372430363052931612313698320861967671070891118800287050683707652563216998263026892800
141637723610785444246127910799497632398083360875263187016998588824084899011108107635578646348859217078824682312959233344350096732542120061725671079362694382211962928924898099200
15119131816542859407559394760796224333553481333025378146874773800816802958830472843322906735939046402777631969935306710504079854827800161439884965184184552847166016101936933195593745898422593198475277107200
Technical details
Node id
n-f47e83b3f1520fc5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5d3f47477444bdf9

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: 1,     4;
   3: 1,    24,        9;
   4: 1,    92,      207,        16;
   5: 1,   304,     2070,      1216,         25;
   6: 1,   948,    15138,     26912,       5925,         36;
   7: 1,  2888,    94887,    376768,     263575,      25992,         49;
   8: 1,  8716,   545913,   4155568,    6493075,    2183652,     106771,        64;
   9: 1, 26208,  2984508,  39764608,  116875350,   89470368,   16248988,    419328,      81;
  10: 1, 78692, 15807060, 347327552, 1735332350, 2498878584, 1063690628, 112405760, 1593513, 100;
%F Formula
The row polynomials satisfy (4*n^2-12*n+9)*P(n,x) = (4*n^2-8*n+3-(32*n^3-144*n^2+152*n+20)*x+(144*n^4-896*n^3+1976*n^2-1824*n+585)*x^2)*P(n-2,x) - ((16*n^2-32*n)*x-(32*n^3-48*n^2+8*n-52)*x^2-(64*n^4-608*n^3+1664*n^2-1752*n+612)*x^3)*d/dx P(n-2,x) - ((16*n^2-32*n+4)*x^2-(64*n^3-224*n^2+256*n-104)*x^3+(64*n^3-240*n^2+288*n-108)*x^4)*d^2/dx^2 P(n-2,x) - (4*n-6-(32*n^3-144*n^2+216*n-108)*x)*P(n-1,x) + ((16*n^2-40*n+20)*x-(32*n^3-112*n^2+120*n-36)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 8*n + 3 - (32*n^3 - 144*n^2 + 152*n + 20)*x + (144*n^4 - 896*n^3 + 1976*n^2 - 1824*n + 585)*x^2)*P[n - 2, x] - ((16*n^2 - 32*n)*x - (32*n^3 - 48*n^2 + 8*n - 52)*x^2 - (64*n^4 - 608*n^3 + 1664*n^2 - 1752*n + 612)*x^3)*D[P[n - 2, x], x] - ((16*n^2 - 32*n + 4)*x^2 - (64*n^3 - 224*n^2 + 256*n - 104)*x^3 + (64*n^3 - 240*n^2 + 288*n - 108)*x^4)*D[P[n - 2, x], {x, 2}] - (4*n - 6 - (32*n^3 - 144*n^2 + 216*n - 108)*x)*P[n - 1, x] + ((16*n^2 - 40*n + 20)*x - (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 12*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 8*n + 3 - (32*n^3 - 144*n^2 + 152*n + 20)*x + (144*n^4 - 896*n^3 + 1976*n^2 - 1824*n + 585)*x^2)*v[i - 2] - ((16*n^2 - 32*n)*x - (32*n^3 - 48*n^2 + 8*n - 52)*x^2 - (64*n^4 - 608*n^3 + 1664*n^2 - 1752*n + 612)*x^3)*deriv(v[i - 2], x) - ((16*n^2 - 32*n + 4)*x^2 - (64*n^3 - 224*n^2 + 256*n - 104)*x^3 + (64*n^3 - 240*n^2 + 288*n - 108)*x^4)*deriv(deriv(v[i - 2], x), x) - (4*n - 6 - (32*n^3 - 144*n^2 + 216*n - 108)*x)*v[i - 1] + ((16*n^2 - 40*n + 20)*x - (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 12*n + 9)); 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.