Real-rooted atlas

A304236 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-12\,n - 5\,n x + 12\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(30 - 12\,x - 114\,x^{2} - 72\,x^{3} - 69\,n - 89\,n x + 54\,n x^{2} + 72\,n x^{3} + 13\,n^{2} + 90\,n^{2} x + 72\,n^{2} x^{2} + 21\,n^{3} + 18\,n^{3} x\right)\,P_{n-1}(x) + \left(-91 - 422\,x - 715\,x^{2} - 528\,x^{3} - 144\,x^{4} - 130\,n - 406\,n x - 420\,n x^{2} - 144\,n x^{3} - 39\,n^{2} - 75\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
27310
313619
446459100
510612027253
6301474216271018
773913507381082935
81933428732139188110432
948011212010719378040531825
10121243505537512179553645243106534
11300049674411874667932174151458332299
127442526660437486825903889505138517291079020
1318291771838011306889013683759217484451033389221
14448819192211533646863074760154791041334351030218710824850
Technical details
Node id
n-a589983e2eca5f67 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7ac04ff230617c4a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:    7,     3;
  3:   13,     6;
  4:   46,    45,     9;
  5:  106,   120,    27;
  6:  301,   474,   216,   27;
  7:  739,  1350,   738,  108;
  8: 1933,  4287,  3213,  918,  81;
  9: 4801, 12120, 10719, 3780, 405;
%F Formula
The row polynomials satisfy (3*n^3+4*n^2-12*n+(12*n^2-5*n)*x+12*n*x^2)*P(n,x) = (21*n^3+13*n^2-69*n+30+(18*n^3+90*n^2-89*n-12)*x+(72*n^2+54*n-114)*x^2+(72*n-72)*x^3)*P(n-1,x) - (39*n^2+130*n+91+(75*n^2+406*n+422)*x+(36*n^2+420*n+715)*x^2+(144*n+528)*x^3+144*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((21*n^3 + 13*n^2 - 69*n + 30 + (18*n^3 + 90*n^2 - 89*n - 12)*x + (72*n^2 + 54*n - 114)*x^2 + (72*n - 72)*x^3)*P[n - 1, x] - (39*n^2 + 130*n + 91 + (75*n^2 + 406*n + 422)*x + (36*n^2 + 420*n + 715)*x^2 + (144*n + 528)*x^3 + 144*x^4)*D[P[n - 1, x], x])/(3*n^3 + 4*n^2 - 12*n + (12*n^2 - 5*n)*x + 12*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((21*n^3 + 13*n^2 - 69*n + 30 + (18*n^3 + 90*n^2 - 89*n - 12)*x + (72*n^2 + 54*n - 114)*x^2 + (72*n - 72)*x^3)*v[i - 1] - (39*n^2 + 130*n + 91 + (75*n^2 + 406*n + 422)*x + (36*n^2 + 420*n + 715)*x^2 + (144*n + 528)*x^3 + 144*x^4)*deriv(v[i - 1], x))/(3*n^3 + 4*n^2 - 12*n + (12*n^2 - 5*n)*x + 12*n*x^2)); 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. [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.