Real-rooted atlas

A144439 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-4\,x^{2} - 8\,n x^{2}\right)\,P_{n-4}(x) + \left(-4\,x + 12\,x^{2} - 8\,n x + 32\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-4\,x^{2} + 4\,x^{3} - 8\,n x^{2} + 8\,n x^{3}\right)\,P_{n-3}'(x) + \left(-3 + 6\,x + 5\,x^{2} - 6\,n + 24\,n x - 22\,n x^{2} - 8\,n^{2} x + 28\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 2\,x^{3} - 4\,n x + 12\,n x^{2} - 8\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 6\,x + 8\,n - 16\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 8 + 2\,x\)
\(P_{3}(x) = 31 + 62\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28210
33162396
410265230641064
5317481272181268513620
6964301741030086034848206199320
72907173346114284115237884333651744273293696
88738946168109506182770287218251030283850461166860759104
9262335004608956246404129824885162281101912492801751612820986491238841360
107872025949914784814664540353780011426275600810530670018312342161037996567084801027681705760
1123618313284586261704392796458313252021452897707025743477248411302048191016454504744597806379236183011672925558272
12708574674463876470570317627232043221083584261327020672072714602450179658578281446408644216141171095286337231938077943141217684850710400
132125749340585851635104337201477174048750205495642427331515240102632964017780119738458087930278837304173641609687951170144573380187322218382550898813499713846329152
1463772761713841432625768291145287941134370960478961704359346031109266246457945349717618211456414790216619439214213106784682281985283592740109448373419936102830485956829045881415109559094813568
Technical details
Node id
n-32fce1bd1e1cbfb9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ec24c0ec2e60f511

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     1;
  2:     8,       2;
  3:    31,      62,        3;
  4:   102,     652,      306,         4;
  5:   317,    4812,     7218,      1268,         5;
  6:   964,   30174,   103008,     60348,      4820,         6;
  7:  2907,  173346,  1142841,   1523788,    433365,     17442,        7;
  8:  8738,  946168, 10950618,  27702872,  18251030,   2838504,    61166,      8;
  9: 26233, 5004608, 95624640, 412982488, 516228110, 191249280, 17516128, 209864, 9;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = -(8*n+4)*x^2*P(n-4,x) - ((8*n+4)*x+(16*n^2-32*n-12)*x^2)*P(n-3,x) - ((8*n+4)*x^2-(8*n+4)*x^3)*d/dx P(n-3,x) - (6*n+3+(8*n^2-24*n-6)*x+(8*n^3-28*n^2+22*n-5)*x^2)*P(n-2,x) - ((4*n+2)*x+(8*n^2-12*n)*x^2-(8*n^2-8*n+2)*x^3)*d/dx P(n-2,x) + (8*n-2+(8*n^2-16*n+6)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 31 + 62*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 8 + 2*x;
P[3, x_] := 31 + 62*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 4)*x^2*P[n - 4, x] - ((8*n + 4)*x + (16*n^2 - 32*n - 12)*x^2)*P[n - 3, x] - ((8*n + 4)*x^2 - (8*n + 4)*x^3)*D[P[n - 3, x], x] - (6*n + 3 + (8*n^2 - 24*n - 6)*x + (8*n^3 - 28*n^2 + 22*n - 5)*x^2)*P[n - 2, x] - ((4*n + 2)*x + (8*n^2 - 12*n)*x^2 - (8*n^2 - 8*n + 2)*x^3)*D[P[n - 2, x], x] + (8*n - 2 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*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] = 0; v[2] = 1; v[3] = 8 + 2*x; v[4] = 31 + 62*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n + 4)*x^2*v[i - 4] - ((8*n + 4)*x + (16*n^2 - 32*n - 12)*x^2)*v[i - 3] - ((8*n + 4)*x^2 - (8*n + 4)*x^3)*deriv(v[i - 3], x) - (6*n + 3 + (8*n^2 - 24*n - 6)*x + (8*n^3 - 28*n^2 + 22*n - 5)*x^2)*v[i - 2] - ((4*n + 2)*x + (8*n^2 - 12*n)*x^2 - (8*n^2 - 8*n + 2)*x^3)*deriv(v[i - 2], x) + (8*n - 2 + (8*n^2 - 16*n + 6)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.