Real-rooted atlas

A257607 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 4\,x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110515
211011025245
3132019809101254335
4171603432023870671062582685
52402406006005610002409004651031251692375
636036001081080012698400737880021979103103101562537015445
7576576002018016002858856002102100008540532018798780201811078125861855135

All 15 rows

Technical details
Node id
n-fa0ad7db8c6303f8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ec2ecf19f26d1b35

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:          10,           5;
  2:         110,         110,           25;
  3:        1320,        1980,          910,          125;
  4:       17160,       34320,        23870,         6710,          625;
  5:      240240,      600600,       561000,       240900,        46510,        3125;
  6:     3603600,    10810800,     12698400,      7378800,      2197910,      310310,      15625;
  7:    57657600,   201801600,    285885600,    210210000,     85405320,    18798780,    2018110,      78125;
  8:   980179200,  3920716800,   6508101600,   5801796000,   3002159160,   908827920,  153790670,   12889910,   390625;
  9: 17643225600, 79394515200, 151178227200, 158616057600, 100048188240, 38834195400, 9116111400, 1218323700, 81245710, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (n+9+(n+4)*x)*P(n-1,x) - (x+x^2)*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[(n + 9 + (n + 4)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
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] = (n + 9 + (n + 4)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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

Same fingerprint ?

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