Real-rooted atlas

A121574 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} - 4\,n - 9\,n x - 4\,n x^{2} + 3\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(24 + 60\,x + 48\,x^{2} + 12\,x^{3} - 13\,n - 29\,n x - 20\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}'(x) + \left(13 + 42\,x + 49\,x^{2} + 24\,x^{3} + 4\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21013427
33364408145
410928126510816779
536011581461902272324185
6118945847245600127446566422483
739271764833522348562140877601536128120785
8129706657114769118485814268169496208483520256648891
9428372472286273859183348540235230182108325392079365123486025
101414819068992590873434232147262763489277176887660779213542417664102418727907
114672803293724104610301975442824635827212974241301903256258801683712332288389122048100611585
1215433211186411241468310871185381224849341213794798691009745292372170461124517376800000849924096540513739
135097243424399821618879083745076925860821336550415565378057053283422221490326724966035211805952189593618432081922903791865
14168350501509218656238761101575885740271686525533811533473131789803219284054511661209444686098881400559363018470444339203973121638415599986803
Technical details
Node id
n-cf4f50cac9609b50 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e1118c67b24d5944

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:     3,      2;
  2:    10,     13,      4;
  3:    33,     64,     40,      8;
  4:   109,    281,    265,    108,     16;
  5:   360,   1158,   1461,    902,    272,     32;
  6:  1189,   4584,   7245,   6001,   2744,    656,     64;
  7:  3927,  17648,  33522,  34856,  21408,   7760,   1536,   128;
  8: 12970,  66571, 147691, 184858, 142681,  69496,  20848,  3520,  256;
  9: 42837, 247228, 627385, 918334, 854023, 523018, 210832, 53920, 7936, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n^2-4*n+4+(3*n^2-9*n+8)*x-(4*n-4)*x^2)*P(n-1,x) - (13*n-24+(29*n-60)*x+(20*n-48)*x^2+(4*n-12)*x^3)*d/dx P(n-1,x) + (13+42*x+49*x^2+24*x^3+4*x^4)*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[((3*n^2 - 4*n + 4 + (3*n^2 - 9*n + 8)*x - (4*n - 4)*x^2)*P[n - 1, x] - (13*n - 24 + (29*n - 60)*x + (20*n - 48)*x^2 + (4*n - 12)*x^3)*D[P[n - 1, x], x] + (13 + 42*x + 49*x^2 + 24*x^3 + 4*x^4)*D[P[n - 1, x], {x, 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] = ((3*n^2 - 4*n + 4 + (3*n^2 - 9*n + 8)*x - (4*n - 4)*x^2)*v[i - 1] - (13*n - 24 + (29*n - 60)*x + (20*n - 48)*x^2 + (4*n - 12)*x^3)*deriv(v[i - 1], x) + (13 + 42*x + 49*x^2 + 24*x^3 + 4*x^4)*deriv(deriv(v[i - 1], x), x))/(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