Real-rooted atlas

A026729 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-52 + 464\,x + 576\,x^{2} - 256\,x^{3} + 228\,n - 592\,n x - 832\,n x^{2} + 256\,n x^{3} - 240\,n^{2} + 128\,n^{2} x + 256\,n^{2} x^{2} + 64\,n^{3}\right)\,P_{n}(x) = \left(980 - 896\,x - 2944\,x^{2} + 1024\,x^{4} - 2200\,n - 1184\,n x + 1920\,n x^{2} + 512\,n x^{3} + 1056\,n^{2} + 512\,n^{2} x - 512\,n^{2} x^{2} - 128\,n^{3}\right)\,P_{n-1}(x) + \left(105 + 244\,x - 992\,x^{2} - 896\,x^{3} + 1280\,x^{4} + 1024\,x^{5} - 220\,n - 896\,n x + 384\,n x^{2} + 2048\,n x^{3} + 1024\,n x^{4} + 80\,n^{2} + 384\,n^{2} x + 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2549
314441674
43118827264555
565572160014722563965
612814966256110727424102427400
72453560199045318468096358404096184925
8457795655792200640388864388096167936163841226125

All 15 rows

Technical details
Node id
n-28995e0481765710 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-176531bdf5bff709

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    5,     4;
   3:   14,    44,     16;
   4:   31,   188,    272,      64;
   5:   65,   572,   1600,    1472,     256;
   6:  128,  1496,   6256,   11072,    7424,     1024;
   7:  245,  3560,  19904,   53184,   68096,    35840,     4096;
   8:  457,  7956,  55792,  200640,  388864,   388096,   167936,    16384;
   9:  838, 16996, 143456,  650304, 1702656,  2571264,  2097152,   770048,   65536;
  10: 1515, 35092, 346480, 1897792, 6283776, 12840960, 15822848, 10895360, 3473408, 262144;
%F Formula
The row polynomials satisfy (64*n^3-240*n^2+228*n-52+(128*n^2-592*n+464)*x+(256*n^2-832*n+576)*x^2+(256*n-256)*x^3)*P(n,x) = -(128*n^3-1056*n^2+2200*n-980-(512*n^2-1184*n-896)*x+(512*n^2-1920*n+2944)*x^2-512*n*x^3-1024*x^4)*P(n-1,x) + (80*n^2-220*n+105+(384*n^2-896*n+244)*x+(256*n^2+384*n-992)*x^2+(2048*n-896)*x^3+(1024*n+1280)*x^4+1024*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(128*n^3 - 1056*n^2 + 2200*n - 980 - (512*n^2 - 1184*n - 896)*x + (512*n^2 - 1920*n + 2944)*x^2 - 512*n*x^3 - 1024*x^4)*P[n - 1, x] + (80*n^2 - 220*n + 105 + (384*n^2 - 896*n + 244)*x + (256*n^2 + 384*n - 992)*x^2 + (2048*n - 896)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*D[P[n - 1, x], x])/(64*n^3 - 240*n^2 + 228*n - 52 + (128*n^2 - 592*n + 464)*x + (256*n^2 - 832*n + 576)*x^2 + (256*n - 256)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(128*n^3 - 1056*n^2 + 2200*n - 980 - (512*n^2 - 1184*n - 896)*x + (512*n^2 - 1920*n + 2944)*x^2 - 512*n*x^3 - 1024*x^4)*v[i - 1] + (80*n^2 - 220*n + 105 + (384*n^2 - 896*n + 244)*x + (256*n^2 + 384*n - 992)*x^2 + (2048*n - 896)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*deriv(v[i - 1], x))/(64*n^3 - 240*n^2 + 228*n - 52 + (128*n^2 - 592*n + 464)*x + (256*n^2 - 832*n + 576)*x^2 + (256*n - 256)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.