Real-rooted atlas

A144436 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
21010121
34872261147
42805603385811237
51872468039841296122111955
61422442672469122270444902501131253
71212804244805721603692001144481471250611616787
811484164593664733481659266242527016535600466581018122113813
9119692805386176099414400965955205265643215829720238990414505620421332864115
101362557446812787201427199360162112512010785782244231673769342715210330544445370409015471811701
11168294604892562032642169788544028218960000221946345601078616380831724456645289283204369204813565528186197583223891
12224233216001345399296003489288880645113617523204645211644162698533693449977664441622623274688290620549618199224041105781637811877120669141
13320658286592208427886284859254421667849662864425984996048354560067412939973123012874547968871604929664155467362576156211830807498702241241361632762138751351625011
1448995675852803429697309696010606024865587219050084167475221976035935436816995527356416089266036962560315654354974727291623966416103888406488082599677792306564318437404650655301854720947213877
Technical details
Node id
n-889768c2c30f1bc9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1c1d701204aa53d8

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:        2,        1;
  2:       10,       10,        1;
  3:       48,       72,       26,        1;
  4:      280,      560,      338,       58,        1;
  5:     1872,     4680,     3984,     1296,      122,        1;
  6:    14224,    42672,    46912,    22704,     4490,      250,       1;
  7:   121280,   424480,   572160,   369200,   114448,    14712,     506,      1;
  8:  1148416,  4593664,  7334816,  5926624,  2527016,   535600,   46658,   1018,    1;
  9: 11969280, 53861760, 99414400, 96595520, 52656432, 15829720, 2389904, 145056, 2042, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+4*x)*P(n-2,x) + (n+1+n*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(4 + 4*x)*P[n - 2, x] + (n + 1 + n*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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 4*x)*v[i - 2] + (n + 1 + n*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