Real-rooted atlas

A113413 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-2}(x) + \left(3 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21416434
35088488194
4178424360128161106
5634190422001216320326306
62258818411976902436807686435954
7804234144605045785632160103681792128204994
8286421394002901203365762376001043202777640962561168786
91020105598561338456182668815659848732163162887168092165126663906
1036331422196725994120941036894996806435456295948891136017971220480102437994674
1112939628709376262183284654963254105120431841282412838494289922523648440320450562048216628994
124608514338820241125013202228945282932347202696328961775432968428544028611072676864010588169830440961235123666
131641346613086977647512341610392221441526802720158994662412065207046750576642783093768345600017684480250675221299281927042134306
145845742650241733619798764884738704960769111094489491805447693807744496187187224108364808777461762356203524520345658572804587521638440151166194
Technical details
Node id
n-2c1debc2d4a26248 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-d33ca026027e399a

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:     14,     16,       4;
  3:     50,     88,      48,       8;
  4:    178,    424,     360,     128,      16;
  5:    634,   1904,    2200,    1216,     320,     32;
  6:   2258,   8184,   11976,    9024,    3680,    768,     64;
  7:   8042,  34144,   60504,   57856,   32160,  10368,   1792,   128;
  8:  28642, 139400,  290120,  336576,  237600, 104320,  27776,  4096,  256;
  9: 102010, 559856, 1338456, 1826688, 1565984, 873216, 316288, 71680, 9216, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + (3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + (3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + (3 + 2*x)*v[i - 1]); 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