Real-rooted atlas

A059473 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 4\,x^{2} + 3\,n - 3\,n x - 4\,n x^{2} + 3\,n^{2} x\right)\,P_{n}(x) = \left(-21 - 78\,x - 92\,x^{2} - 32\,x^{3} + 42\,n + 102\,n x + 52\,n x^{2} - 9\,n^{2} - 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 48\,x - 90\,x^{2} - 68\,x^{3} - 16\,x^{4} + 9\,n + 36\,n x + 42\,n x^{2} + 12\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2363268
3240448192880
4142440643648102410160
579203046442048245765120110128
642288204736382656344064148480245761146800
7219520128153630259203690496244736083558411468811615104

All 15 rows

Technical details
Node id
n-c684e1d0e7e1e22c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-daa7ccce45bcf643

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:       4;
  2:      36,       32;
  3:     240,      448,       192;
  4:    1424,     4064,      3648,      1024;
  5:    7920,    30464,     42048,     24576,      5120;
  6:   42288,   204736,    382656,    344064,    148480,     24576;
  7:  219520,  1281536,   3025920,   3690496,   2447360,    835584,   114688;
  8: 1116288,  7630080,  21786624,  33629184,  30259200,  15851520,  4472832,   524288;
  9: 5587776, 43769856, 146668032, 274231296, 312529920, 222068736, 95993856, 23068672, 2359296;
%F Formula
The row polynomials satisfy (3*n-3+(3*n^2-3*n)*x-(4*n-4)*x^2)*P(n,x) = -(9*n^2-42*n+21+(12*n^2-102*n+78)*x-(52*n-92)*x^2+32*x^3)*P(n-1,x) + (9*n-9+(36*n-48)*x+(42*n-90)*x^2+(12*n-68)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 42*n + 21 + (12*n^2 - 102*n + 78)*x - (52*n - 92)*x^2 + 32*x^3)*P[n - 1, x] + (9*n - 9 + (36*n - 48)*x + (42*n - 90)*x^2 + (12*n - 68)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(3*n - 3 + (3*n^2 - 3*n)*x - (4*n - 4)*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 42*n + 21 + (12*n^2 - 102*n + 78)*x - (52*n - 92)*x^2 + 32*x^3)*v[i - 1] + (9*n - 9 + (36*n - 48)*x + (42*n - 90)*x^2 + (12*n - 68)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(3*n - 3 + (3*n^2 - 3*n)*x - (4*n - 4)*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.