Real-rooted atlas

A081578 → basis A013609 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(13 + 4\,x - 13\,n - 2\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(71 + 59\,x + 8\,x^{2} - 26\,n - 20\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(39 + 56\,x + 23\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
22522451
3136180728396
47691340816200163141
543009334767229365123224786
6241216261064740337849264124864195831
71351844084885092323370721268482688029441281546776

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(9 + 6\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
Technical details
Node id
n-d04147815db6454e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-bb97ff5ef4c0e04f)
Fingerprint
f6-a5ecc6fad53bda4b

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:       4,        2;
  2:      25,       22,        4;
  3:     136,      180,       72,        8;
  4:     769,     1340,      816,      200,       16;
  5:    4300,     9334,     7672,     2936,      512,      32;
  6:   24121,    62610,    64740,    33784,     9264,    1248,      64;
  7:  135184,   408488,   509232,   337072,   126848,   26880,    2944,    128;
  8:  757825,  2612536,  3812224,  3059248,  1467616,  428032,   73600,   6784,   256;
  9: 4247956, 16453290, 27507984, 25950480, 15153024, 5650368, 1338240, 193152, 15360, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n^2-13*n+13-(2*n-4)*x)*P(n-1,x) - (26*n-71+(20*n-59)*x+(2*n-8)*x^2)*d/dx P(n-1,x) + (39+56*x+23*x^2+2*x^3)*d^2/dx^2 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^2 - 13*n + 13 - (2*n - 4)*x)*P[n - 1, x] - (26*n - 71 + (20*n - 59)*x + (2*n - 8)*x^2)*D[P[n - 1, x], x] + (39 + 56*x + 23*x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}])/(n)]];
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^2 - 13*n + 13 - (2*n - 4)*x)*v[i - 1] - (26*n - 71 + (20*n - 59)*x + (2*n - 8)*x^2)*deriv(v[i - 1], x) + (39 + 56*x + 23*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x))/(n)); 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