Real-rooted atlas

A099040 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n - 2\,n x + 4\,n x^{2} + 2\,n x^{3} + 3\,n^{2} + 6\,n^{2} x + 2\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-12 + 28\,x^{2} + 20\,x^{3} + 4\,x^{4} + 8\,n + 22\,n x + 8\,n x^{2} + 6\,n^{2} - 6\,n^{2} x - 4\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(12 + 64\,x + 116\,x^{2} + 92\,x^{3} + 32\,x^{4} + 4\,x^{5} + 18\,n + 60\,n x + 66\,n x^{2} + 28\,n x^{3} + 4\,n x^{4} + 6\,n^{2} + 8\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21618438
356104568224
4188488436152161280
5608204025761496384327136
619207912129761069645769286439072
7595229120588806323238464129922176128210944
8181921031042481603293442614561256323500849922561126144
9549763543369899521566784153939296617638297690752112645125957120
10164608118928037845126962752816844863507523289280110835222835225088102431272448
1148908839156481398515229341312400561923721638423889536105189123080192561152552962048163110912
12144377612688256502686081184874241846244481997600001533182088368806432018944828620813527041208324096846061568
1342383364056665617657267246204838480946227210002976008945297925840085762770165769364787221704704320921626214481924367572992
14123822081282202886082741761749954432340571289647346297604838106880368157504020890777608755456002650091525560115275120645652481638422452183040
Technical details
Node id
n-2f41a62f7aaca14e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-fdb1bb4a756fa5c9

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,     18,      4;
  3:    56,    104,     56,       8;
  4:   188,    488,    436,     152,      16;
  5:   608,   2040,   2576,    1496,     384,     32;
  6:  1920,   7912,  12976,   10696,    4576,    928,     64;
  7:  5952,  29120,  58880,   63232,   38464,  12992,   2176,   128;
  8: 18192, 103104, 248160,  329344,  261456, 125632,  35008,  4992,   256;
  9: 54976, 354336, 989952, 1566784, 1539392, 966176, 382976, 90752, 11264, 512;
%F Formula
The row polynomials satisfy (n^3+3*n^2-4*n+(6*n^2-2*n)*x+(2*n^2+4*n)*x^2+2*n*x^3)*P(n,x) = -(2*n^3-6*n^2-8*n+12+(6*n^2-22*n)*x+(4*n^2-8*n-28)*x^2-20*x^3-4*x^4)*P(n-1,x) + (6*n^2+18*n+12+(8*n^2+60*n+64)*x+(2*n^2+66*n+116)*x^2+(28*n+92)*x^3+(4*n+32)*x^4+4*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[(-(2*n^3 - 6*n^2 - 8*n + 12 + (6*n^2 - 22*n)*x + (4*n^2 - 8*n - 28)*x^2 - 20*x^3 - 4*x^4)*P[n - 1, x] + (6*n^2 + 18*n + 12 + (8*n^2 + 60*n + 64)*x + (2*n^2 + 66*n + 116)*x^2 + (28*n + 92)*x^3 + (4*n + 32)*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n^3 + 3*n^2 - 4*n + (6*n^2 - 2*n)*x + (2*n^2 + 4*n)*x^2 + 2*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] = (-(2*n^3 - 6*n^2 - 8*n + 12 + (6*n^2 - 22*n)*x + (4*n^2 - 8*n - 28)*x^2 - 20*x^3 - 4*x^4)*v[i - 1] + (6*n^2 + 18*n + 12 + (8*n^2 + 60*n + 64)*x + (2*n^2 + 66*n + 116)*x^2 + (28*n + 92)*x^3 + (4*n + 32)*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n^3 + 3*n^2 - 4*n + (6*n^2 - 2*n)*x + (2*n^2 + 4*n)*x^2 + 2*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

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
  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.