Real-rooted atlas

A113413 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(8 + 5\,x + x^{2} - 12\,n - 4\,n x + 4\,n^{2} - 2\,n^{2} x - n^{2} x^{2} + n^{3} x\right)\,P_{n}(x) = \left(16 + 16\,x + 2\,x^{2} - 56\,n - 75\,n x - 27\,n x^{2} - 3\,n x^{3} + 32\,n^{2} + 35\,n^{2} x + 7\,n^{2} x^{2} - 4\,n^{3} - 3\,n^{3} x\right)\,P_{n-1}(x) + \left(8 + 16\,x + 9\,x^{2} + x^{3} - 24\,n - 56\,n x - 43\,n x^{2} - 12\,n x^{3} - n x^{4} + 8\,n^{2} + 16\,n^{2} x + 9\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2628
32318344
47696364212
5233408246605952
6682153012845009064092
71935529857213100885126717072
8536817360229681604063601428168869700
914641546248564473432375301167621562169279928
103940616662630223230772019326077448197683096270101109836
1110493549597010219831206040900490443520145670314644275330114354688
12276996144724833400924485420388644022769889189042550964770057203961216941012
13725849415432810616994159929801578769510741812515740417583124220556952074584681365434888
14189025811761594329764445508082861059830473695742642231210716072315689466671098076951654614251208668
Technical details
Node id
n-c85d73af77ec3683 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6f17d48b11ead97b

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:    23,    18,     3;
  4:    76,    96,    36,     4;
  5:   233,   408,   246,    60,     5;
  6:   682,  1530,  1284,   500,    90,     6;
  7:  1935,  5298,  5721,  3100,   885,   126,    7;
  8:  5368, 17360, 22968, 16040,  6360,  1428,  168,   8;
  9: 14641, 54624, 85644, 73432, 37530, 11676, 2156, 216, 9;
%F Formula
The row polynomials satisfy (4*n^2-12*n+8+(n^3-2*n^2-4*n+5)*x-(n^2-1)*x^2)*P(n,x) = -(4*n^3-32*n^2+56*n-16+(3*n^3-35*n^2+75*n-16)*x-(7*n^2-27*n+2)*x^2+3*n*x^3)*P(n-1,x) + (8*n^2-24*n+8+(16*n^2-56*n+16)*x+(9*n^2-43*n+9)*x^2+(n^2-12*n+1)*x^3-n*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[(-(4*n^3 - 32*n^2 + 56*n - 16 + (3*n^3 - 35*n^2 + 75*n - 16)*x - (7*n^2 - 27*n + 2)*x^2 + 3*n*x^3)*P[n - 1, x] + (8*n^2 - 24*n + 8 + (16*n^2 - 56*n + 16)*x + (9*n^2 - 43*n + 9)*x^2 + (n^2 - 12*n + 1)*x^3 - n*x^4)*D[P[n - 1, x], x])/(4*n^2 - 12*n + 8 + (n^3 - 2*n^2 - 4*n + 5)*x - (n^2 - 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] = (-(4*n^3 - 32*n^2 + 56*n - 16 + (3*n^3 - 35*n^2 + 75*n - 16)*x - (7*n^2 - 27*n + 2)*x^2 + 3*n*x^3)*v[i - 1] + (8*n^2 - 24*n + 8 + (16*n^2 - 56*n + 16)*x + (9*n^2 - 43*n + 9)*x^2 + (n^2 - 12*n + 1)*x^3 - n*x^4)*deriv(v[i - 1], x))/(4*n^2 - 12*n + 8 + (n^3 - 2*n^2 - 4*n + 5)*x - (n^2 - 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.