Real-rooted atlas

A081577 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + x^{2} + 6\,n - 3\,n x - n x^{2} + 3\,n^{2} x\right)\,P_{n}(x) = \left(-48 - 66\,x - 22\,x^{2} - 2\,x^{3} + 78\,n + 87\,n x + 14\,n x^{2} - 18\,n^{2} - 15\,n^{2} x\right)\,P_{n-1}(x) + \left(-36 - 84\,x - 63\,x^{2} - 16\,x^{3} - x^{4} + 36\,n + 72\,n x + 39\,n x^{2} + 3\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2628
32420347
488116424250
53005443247251245
69842272189669611065964
731368800948049281280156727787
897923235242768289441068021242108126878
930096114432179424150048724202044832762729570425
1091360392928713088711264426120157680357844784342102533360
11274560131782427177123151872226128010414083097925849666964201111140071
128183044336064100171201325145611092800613440022740485624009064890605061248586818
13242195214042624359272325341004851152160330900481475443245488009590041345601192460013210453397
147125888448744961259608322079383042243811201664542088697561632282432848210415542401928081533670214906238100
Technical details
Node id
n-a138b21fd7d4ae30 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4d47c57260d8ac77

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:     6,      2;
  3:    24,     20,      3;
  4:    88,    116,     42,      4;
  5:   300,    544,    324,     72,     5;
  6:   984,   2272,   1896,    696,   110,     6;
  7:  3136,   8800,   9480,   4928,  1280,   156,    7;
  8:  9792,  32352,  42768,  28944, 10680,  2124,  210,   8;
  9: 30096, 114432, 179424, 150048, 72420, 20448, 3276, 272, 9;
%F Formula
The row polynomials satisfy (6*n-6+(3*n^2-3*n)*x-(n-1)*x^2)*P(n,x) = -(18*n^2-78*n+48+(15*n^2-87*n+66)*x-(14*n-22)*x^2+2*x^3)*P(n-1,x) + (36*n-36+(72*n-84)*x+(39*n-63)*x^2+(3*n-16)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 - 78*n + 48 + (15*n^2 - 87*n + 66)*x - (14*n - 22)*x^2 + 2*x^3)*P[n - 1, x] + (36*n - 36 + (72*n - 84)*x + (39*n - 63)*x^2 + (3*n - 16)*x^3 - x^4)*D[P[n - 1, x], x])/(6*n - 6 + (3*n^2 - 3*n)*x - (n - 1)*x^2)]];
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; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 - 78*n + 48 + (15*n^2 - 87*n + 66)*x - (14*n - 22)*x^2 + 2*x^3)*v[i - 1] + (36*n - 36 + (72*n - 84)*x + (39*n - 63)*x^2 + (3*n - 16)*x^3 - x^4)*deriv(v[i - 1], x))/(6*n - 6 + (3*n^2 - 3*n)*x - (n - 1)*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

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.