Real-rooted atlas

A142963 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(24\,x + 60\,x^{2} + 32\,n x + 56\,n x^{2} - 32\,n^{2} x - 496\,n^{2} x^{2} + 288\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 16\,x^{2} + 80\,x^{3} - 4\,n x + 32\,n x^{2} + 128\,n x^{3} + 32\,n^{2} x^{2} - 640\,n^{2} x^{3} + 128\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} + 16\,x^{4} - 2\,n x^{2} + 32\,n x^{4} + 32\,n^{2} x^{3} - 128\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-1 + 8\,x + 2\,n - 32\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(8\,x^{2} + 4\,n x - 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
21403677
31120648128897
412964644742440012765
5166423256112256716001152208929
611416966601027840211200062553631363866589
712936360072720448033044000341660165130496819279916193
81599212519724290073636784600087116313650095091240271872207361824411357
911212041603762294718723313942800147615471362010715392068652072963060011525120045587547873
10124392134083081139774208258920088001935549907205024079822724225953300488964179020822671872001239041237512620061
111489524231044053695545601828869600002130726753792931275254246415192476835840829463711616011293606400001646114201629491236264714199137
1219808813155728424336523520119969238600020685077194752140593320139776389138908151808420755346961920154657704704000138463363061761175442554886922241141018398971037
13119637640482396010716903795274434430140001828474023870721828793893097472788754564710400014621884596509184108966958684160002772211217053696166193376362496827716464640160563238364550736072481
141392968123656418046178175564844250183185600150438961604851221263328706595136134759227926773760391146510318598656506104565009100800267824359401691136481825669095751681961224244961280575975361740836864001372796645092546653
Technical details
Node id
n-8601e37f53e74b20 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-23668f2803585b2c

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: 1,     8;
  2: 1,    40,      36;
  3: 1,   120,     648,       128;
  4: 1,   296,    4644,      7424,        400;
  5: 1,   664,   23256,    112256,      71600,        1152;
  6: 1,  1416,   96660,   1027840,    2112000,      625536,        3136;
  7: 1,  2936,  360072,   7204480,   33044000,    34166016,     5130496,       8192;
  8: 1,  5992, 1251972,  42900736,  367846000,   871163136,   500950912,   40271872,     20736;
  9: 1, 12120, 4160376, 229471872, 3313942800, 14761547136, 20107153920, 6865207296, 306001152, 51200;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = -((32*n^2-32*n-24)*x-(288*n^3-496*n^2+56*n+60)*x^2)*P(n-2,x) - ((4*n+2)*x-(32*n^2+32*n+16)*x^2-(128*n^3-640*n^2+128*n+80)*x^3)*d/dx P(n-2,x) - ((2*n+1)*x^2-32*n^2*x^3+(128*n^2-32*n-16)*x^4)*d^2/dx^2 P(n-2,x) + (2*n-1+(32*n^2-32*n+8)*x)*P(n-1,x) + (4*n*x-(32*n^2-8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((32*n^2 - 32*n - 24)*x - (288*n^3 - 496*n^2 + 56*n + 60)*x^2)*P[n - 2, x] - ((4*n + 2)*x - (32*n^2 + 32*n + 16)*x^2 - (128*n^3 - 640*n^2 + 128*n + 80)*x^3)*D[P[n - 2, x], x] - ((2*n + 1)*x^2 - 32*n^2*x^3 + (128*n^2 - 32*n - 16)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n - 1 + (32*n^2 - 32*n + 8)*x)*P[n - 1, x] + (4*n*x - (32*n^2 - 8)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-((32*n^2 - 32*n - 24)*x - (288*n^3 - 496*n^2 + 56*n + 60)*x^2)*v[i - 2] - ((4*n + 2)*x - (32*n^2 + 32*n + 16)*x^2 - (128*n^3 - 640*n^2 + 128*n + 80)*x^3)*deriv(v[i - 2], x) - ((2*n + 1)*x^2 - 32*n^2*x^3 + (128*n^2 - 32*n - 16)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n - 1 + (32*n^2 - 32*n + 8)*x)*v[i - 1] + (4*n*x - (32*n^2 - 8)*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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.