Real-rooted atlas

A035607 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 15\,x + 11\,x^{2} - 8\,x^{3} - 4\,x^{4} - 2\,n - 12\,n x - 24\,n x^{2} - n x^{3} + 2\,n x^{4} - 2\,n^{2} x + 8\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-4 - 15\,x - 11\,x^{2} + 8\,x^{3} + 4\,x^{4} + 2\,n + 20\,n x + 26\,n x^{2} - 9\,n x^{3} - 6\,n x^{4} + 8\,n^{2} + 4\,n^{2} x - 20\,n^{2} x^{2} - 6\,n^{2} x^{3} - 4\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x + 38\,x^{2} + 87\,x^{3} + 85\,x^{4} + 32\,x^{5} + 4\,x^{6} + 4\,n + 32\,n x + 86\,n x^{2} + 92\,n x^{3} + 34\,n x^{4} + 4\,n x^{5} + 4\,n^{2} + 16\,n^{2} x + 18\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
27411
32332863
47316810416361
5217704744288321985
66232620397626407366410659
71743902417968172808256179212856191

All 15 rows

Technical details
Node id
n-712d4076e029103f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-38a0582a03ef8145

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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,     4;
  3:    23,    32,      8;
  4:    73,   168,    104,     16;
  5:   217,   704,    744,    288,     32;
  6:   623,  2620,   3976,   2640,    736,     64;
  7:  1743,  9024,  17968,  17280,   8256,   1792,   128;
  8:  4785, 29456,  72816,  92512,  64192,  23808,  4224,  256;
  9: 12945, 92416, 273328, 433472, 398720, 214784, 64896, 9728, 512;
%F Formula
The row polynomials satisfy -(2*n-4-(2*n^3-2*n^2-12*n+15)*x-(8*n^2-24*n+11)*x^2-(2*n^2-n-8)*x^3-(2*n-4)*x^4)*P(n,x) = -(4*n^3-8*n^2-2*n+4+(6*n^3-4*n^2-20*n+15)*x+(20*n^2-26*n+11)*x^2+(6*n^2+9*n-8)*x^3+(6*n-4)*x^4)*P(n-1,x) + (4*n^2+4*n+(16*n^2+32*n+6)*x+(18*n^2+86*n+38)*x^2+(4*n^2+92*n+87)*x^3+(34*n+85)*x^4+(4*n+32)*x^5+4*x^6)*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[(-(4*n^3 - 8*n^2 - 2*n + 4 + (6*n^3 - 4*n^2 - 20*n + 15)*x + (20*n^2 - 26*n + 11)*x^2 + (6*n^2 + 9*n - 8)*x^3 + (6*n - 4)*x^4)*P[n - 1, x] + (4*n^2 + 4*n + (16*n^2 + 32*n + 6)*x + (18*n^2 + 86*n + 38)*x^2 + (4*n^2 + 92*n + 87)*x^3 + (34*n + 85)*x^4 + (4*n + 32)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(-2*n + 4 + (2*n^3 - 2*n^2 - 12*n + 15)*x + (8*n^2 - 24*n + 11)*x^2 + (2*n^2 - n - 8)*x^3 + (2*n - 4)*x^4)]];
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] = (-(4*n^3 - 8*n^2 - 2*n + 4 + (6*n^3 - 4*n^2 - 20*n + 15)*x + (20*n^2 - 26*n + 11)*x^2 + (6*n^2 + 9*n - 8)*x^3 + (6*n - 4)*x^4)*v[i - 1] + (4*n^2 + 4*n + (16*n^2 + 32*n + 6)*x + (18*n^2 + 86*n + 38)*x^2 + (4*n^2 + 92*n + 87)*x^3 + (34*n + 85)*x^4 + (4*n + 32)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(-2*n + 4 + (2*n^3 - 2*n^2 - 12*n + 15)*x + (8*n^2 - 24*n + 11)*x^2 + (2*n^2 - n - 8)*x^3 + (2*n - 4)*x^4)); 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.