Real-rooted atlas

A185410 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2325
31518437
4105180848377
594521001500360164921
61039528350273001080014883278365
7135135436590529200294000722406048641473277
82027025756756011060280793800028576804636802438412831938737
93445942514594580024972948022004136010509912026107200290400097920256784384561
10654729075310134825060810750006356750400379658664012859560002291856001791360039244851221523937525
1113749310575720201982501592025435001926484560001383501319205970455568014922230400195888000010944384015713281024652667322517
1231623414322518149089959004465252291500614066953500051576272748002713061070720880599202560166852224000164371468806645196806288384204821672312694697
13790585358062549332526343100133698296031300205977766995000198211715301600123217952457600494306446233601240256489472018164565619201361488128004018560000251596804096782133969325801
1421345804667687514388653516737504259041440954300726689561958360078855779306304005662516067208000271294690867200085326236835840016881180091776019389904934400111732042240024237158400100651008819230481907097849485
Technical details
Node id
n-63fcd80b98d85584 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f2b1ecb4ea267ced

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:        1;
  2:        3,         2;
  3:       15,        18,         4;
  4:      105,       180,        84,         8;
  5:      945,      2100,      1500,       360,        16;
  6:    10395,     28350,     27300,     10800,      1488,       32;
  7:   135135,    436590,    529200,    294000,     72240,     6048,      64;
  8:  2027025,   7567560,  11060280,   7938000,   2857680,   463680,   24384,   128;
  9: 34459425, 145945800, 249729480, 220041360, 105099120, 26107200, 2904000, 97920, 256;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(2*n-2)*x)*P(n-1,x) - (2*x+2*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[(2*n - 1 + (2*n - 2)*x)*P[n - 1, x] - (2*x + 2*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] = (2*n - 1 + (2*n - 2)*x)*v[i - 1] - (2*x + 2*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