Real-rooted atlas

delannoy → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(6\,x^{2} + 2\,x^{3} + 8\,n - 2\,n x - 10\,n x^{2} - 3\,n x^{3} + 6\,n^{2} x + 4\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x) + \left(26\,x^{2} + 22\,x^{3} + 4\,x^{4} - 8\,n x - 20\,n x^{2} - 12\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}'(x) + \left(8\,x^{2} + 18\,x^{3} + 8\,x^{4} + x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21620541
3641287212276
4256704640232291861
510243584454426247007012546
640961740828416228489616202816984581
7163848192016384016998499200327685712408570216
8655363768328929281138688856576390784106112157609853844201
926214417039364669440707788865474563843072143628833088042804237825916286
10104857676021762365849641598976457932803303219215851008501132810021281148205741174718721
1141943043355443211691622423409459229935206425698304015138816061335552167909762965632304920138601177893756
12167772161468006405662310401272446976185565184018514083841297440768644554752225731328544643208612352803064334617940956141
136710886463753420826969374726723469312110207959041254844006410219864064603563622425887651847978193921720315522462227221003328078253535205626
1426843545627514634241266679808034699476992631861411848093040640075237621760516234444802630199296099162880002727353600531465920694704805460700195025360916014461
Technical details
Node id
n-d82988eae907efd9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5f821ed5c717f4fd

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:     16,      20,       5;
  3:     64,     128,      72,      12;
  4:    256,     704,     640,     232,      29;
  5:   1024,    3584,    4544,    2624,     700,      70;
  6:   4096,   17408,   28416,   22848,    9616,    2028,     169;
  7:  16384,   81920,  163840,  169984,   99200,   32768,    5712,    408;
  8:  65536,  376832,  892928, 1138688,  856576,  390784,  106112,  15760,   985;
  9: 262144, 1703936, 4669440, 7077888, 6547456, 3843072, 1436288, 330880, 42804, 2378;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (8*n+(6*n^2-2*n)*x+(4*n^2-10*n+6)*x^2+(n^2-3*n+2)*x^3)*P(n-1,x) - (8*n*x+(20*n-26)*x^2+(12*n-22)*x^3+(2*n-4)*x^4)*d/dx P(n-1,x) + (8*x^2+18*x^3+8*x^4+x^5)*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[((8*n + (6*n^2 - 2*n)*x + (4*n^2 - 10*n + 6)*x^2 + (n^2 - 3*n + 2)*x^3)*P[n - 1, x] - (8*n*x + (20*n - 26)*x^2 + (12*n - 22)*x^3 + (2*n - 4)*x^4)*D[P[n - 1, x], x] + (8*x^2 + 18*x^3 + 8*x^4 + x^5)*D[P[n - 1, x], {x, 2}])/(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] = ((8*n + (6*n^2 - 2*n)*x + (4*n^2 - 10*n + 6)*x^2 + (n^2 - 3*n + 2)*x^3)*v[i - 1] - (8*n*x + (20*n - 26)*x^2 + (12*n - 22)*x^3 + (2*n - 4)*x^4)*deriv(v[i - 1], x) + (8*x^2 + 18*x^3 + 8*x^4 + x^5)*deriv(deriv(v[i - 1], x), x))/(2*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

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