Real-rooted atlas

A330798 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n^{2} + 4\,n^{3} + 4\,n^{3} x + 8\,n^{4} x\right)\,P_{n}(x) = \left(6\,x + 6\,x^{2} - 6\,n - 22\,n x - 18\,n x^{2} + 6\,n^{2} - 10\,n^{2} x - 18\,n^{2} x^{2} + 12\,n^{3} + 46\,n^{3} x + 42\,n^{3} x^{2} + 28\,n^{4} x + 36\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-3\,x - 9\,x^{2} - 6\,x^{3} + n + 5\,n x + 10\,n x^{2} + 6\,n x^{3} + n^{2} + 10\,n^{2} x + 39\,n^{2} x^{2} + 30\,n^{2} x^{3} + 2\,n^{3} x + 20\,n^{3} x^{2} + 18\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
26302763
320168336192716
4708402700330013758285
52523960178203432030030982896210
692418018105105280280382200259896699721116395
7343280080576576198016037128003907008217056049612812926744

All 15 rows

Technical details
Node id
n-313fdaaf3d4e08f3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d744f831d9613171

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:     6,      30,       27;
  3:    20,     168,      336,      192;
  4:    70,     840,     2700,     3300,      1375;
  5:   252,    3960,    17820,    34320,     30030,      9828;
  6:   924,   18018,   105105,   280280,    382200,    259896,     69972;
  7:  3432,   80080,   576576,  1980160,   3712800,   3907008,   2170560,    496128;
  8: 12870,  350064,  3007368, 12697776,  30523500,  43953840,  37604952,  17651304,   3504303;
  9: 48620, 1511640, 15116400, 75969600, 223839000, 411863760, 480507720, 346104000, 140604750, 24667500;
%F Formula
The row polynomials satisfy (4*n^3+2*n^2+(8*n^4+4*n^3)*x)*P(n,x) = (12*n^3+6*n^2-6*n+(28*n^4+46*n^3-10*n^2-22*n+6)*x+(36*n^4+42*n^3-18*n^2-18*n+6)*x^2)*P(n-1,x) + (n^2+n+(2*n^3+10*n^2+5*n-3)*x+(20*n^3+39*n^2+10*n-9)*x^2+(18*n^3+30*n^2+6*n-6)*x^3)*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[((12*n^3 + 6*n^2 - 6*n + (28*n^4 + 46*n^3 - 10*n^2 - 22*n + 6)*x + (36*n^4 + 42*n^3 - 18*n^2 - 18*n + 6)*x^2)*P[n - 1, x] + (n^2 + n + (2*n^3 + 10*n^2 + 5*n - 3)*x + (20*n^3 + 39*n^2 + 10*n - 9)*x^2 + (18*n^3 + 30*n^2 + 6*n - 6)*x^3)*D[P[n - 1, x], x])/(4*n^3 + 2*n^2 + (8*n^4 + 4*n^3)*x)]];
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] = ((12*n^3 + 6*n^2 - 6*n + (28*n^4 + 46*n^3 - 10*n^2 - 22*n + 6)*x + (36*n^4 + 42*n^3 - 18*n^2 - 18*n + 6)*x^2)*v[i - 1] + (n^2 + n + (2*n^3 + 10*n^2 + 5*n - 3)*x + (20*n^3 + 39*n^2 + 10*n - 9)*x^2 + (18*n^3 + 30*n^2 + 6*n - 6)*x^3)*deriv(v[i - 1], x))/(4*n^3 + 2*n^2 + (8*n^4 + 4*n^3)*x)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.