Real-rooted atlas

A113953 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-34\,n - 52\,n x + 40\,n x^{2} + 16\,n x^{3} + 12\,n^{2} + 64\,n^{2} x + 16\,n^{2} x^{2} + 16\,n^{3}\right)\,P_{n}(x) = \left(-174 - 272\,x + 240\,x^{2} + 192\,x^{3} + 32\,x^{4} + 28\,n + 456\,n x + 16\,n x^{2} - 32\,n x^{3} + 192\,n^{2} - 160\,n^{2} x - 64\,n^{2} x^{2} - 64\,n^{3}\right)\,P_{n-1}(x) + \left(-9 + 106\,x + 600\,x^{2} + 848\,x^{3} + 304\,x^{4} + 32\,x^{5} + 54\,n + 408\,n x + 736\,n x^{2} + 288\,n x^{3} + 32\,n x^{4} + 72\,n^{2} + 160\,n^{2} x + 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21116431
32778528165
47128435214416867
516789816721264368324401
639526166508748839848966421951
78997158224443503228432115522112128107757
82031188047111214094415641696704316164864256522747
945114775421187251046472393661395230412882944110085122511081
109939118288602292171078429650883226880219353690214421068824576102411965239
1121691286910164982854024081108758414803648129424647296256255769652172854272204856626533
12470316843004386032162741923864235261304704659281924784179222945024699494412656641187844096266437107
13101335160885011375096471831201273318562344027843002984962684840961657346566897203218577408301875225804881921247354721
14217243373719228895036132526272400820208840743808125276691213367234561017471232544626688199783424481525767098368557056163845814135855
Technical details
Node id
n-3ce0157fb820a665 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-42900778d0475859

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:   11,    16,      4;
  3:   27,    78,     52,      8;
  4:   71,   284,    352,    144,     16;
  5:  167,   898,   1672,   1264,    368,     32;
  6:  395,  2616,   6508,   7488,   3984,    896,     64;
  7:  899,  7158,  22444,  35032,  28432,  11552,   2112,   128;
  8: 2031, 18804,  71112, 140944, 156416,  96704,  31616,  4864,   256;
  9: 4511, 47754, 211872, 510464, 723936, 613952, 304128, 82944, 11008, 512;
%F Formula
The row polynomials satisfy (16*n^3+12*n^2-34*n+(64*n^2-52*n)*x+(16*n^2+40*n)*x^2+16*n*x^3)*P(n,x) = -(64*n^3-192*n^2-28*n+174+(160*n^2-456*n+272)*x+(64*n^2-16*n-240)*x^2+(32*n-192)*x^3-32*x^4)*P(n-1,x) + (72*n^2+54*n-9+(160*n^2+408*n+106)*x+(32*n^2+736*n+600)*x^2+(288*n+848)*x^3+(32*n+304)*x^4+32*x^5)*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[Together[(-(64*n^3 - 192*n^2 - 28*n + 174 + (160*n^2 - 456*n + 272)*x + (64*n^2 - 16*n - 240)*x^2 + (32*n - 192)*x^3 - 32*x^4)*P[n - 1, x] + (72*n^2 + 54*n - 9 + (160*n^2 + 408*n + 106)*x + (32*n^2 + 736*n + 600)*x^2 + (288*n + 848)*x^3 + (32*n + 304)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(16*n^3 + 12*n^2 - 34*n + (64*n^2 - 52*n)*x + (16*n^2 + 40*n)*x^2 + 16*n*x^3)]];
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] = (-(64*n^3 - 192*n^2 - 28*n + 174 + (160*n^2 - 456*n + 272)*x + (64*n^2 - 16*n - 240)*x^2 + (32*n - 192)*x^3 - 32*x^4)*v[i - 1] + (72*n^2 + 54*n - 9 + (160*n^2 + 408*n + 106)*x + (32*n^2 + 736*n + 600)*x^2 + (288*n + 848)*x^3 + (32*n + 304)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(16*n^3 + 12*n^2 - 34*n + (64*n^2 - 52*n)*x + (16*n^2 + 40*n)*x^2 + 16*n*x^3)); 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

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.