Real-rooted atlas

A130850 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,x + n\right)\,P_{n}(x) = \left(-2 - 7\,x - 6\,x^{2} + x^{3} + 2\,x^{4} + n + 8\,n x + 11\,n x^{2} + 2\,n x^{3} - 2\,n x^{4} + 2\,n^{2} + n^{2} x - 4\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-x - 3\,x^{2} - x^{3} + 3\,x^{4} + 2\,x^{5} + 3\,n x^{2} + 6\,n x^{3} + 3\,n x^{4} + n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 4\,x - 3\,x^{2} + 2\,x^{3} - n + 3\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 2\,x^{3} - n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
255111
3182610155
48416095191359
548011409303043612891
6324092409660450891769127635
72520084000107520660241986626941341305439
8221760846720128520098481639992482920783526313829439
92177280937440016480800151905607842240224838033525022748520153672179
1023587200113097600226195200244157760153762840570675601204302013305166620910331831308939
11279417600147692160033130944004104112320306012168014047545603905774406245764852264741935222058114096879303
12359251200020756736000516323808007225073856062365343040342646761601194500736025601896323169606442042768056829541071259705544279
13498161664003123888768008536207680001332374883840130840085376083943699840035401766400096415999680162685723201585305540797078101676232820415164407480987
1474101547520050127517440001492928236800025724860761600283438419264002083474673280010349250912000344615439168074864249460010106725356078538987803111381244963181163971110239784076323
Technical details
Node id
n-9a299b4f34a9f200 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b222bc04a25fc530

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:       2,       1;
  2:       5,       5,        1;
  3:      18,      26,       10,        1;
  4:      84,     160,       95,       19,       1;
  5:     480,    1140,      930,      304,      36,       1;
  6:    3240,    9240,     9660,     4508,     917,      69,      1;
  7:   25200,   84000,   107520,    66024,   19866,    2694,    134,     1;
  8:  221760,  846720,  1285200,   984816,  399924,   82920,   7835,   263,   1;
  9: 2177280, 9374400, 16480800, 15190560, 7842240, 2248380, 335250, 22748, 520, 1;
%F Formula
The row polynomials satisfy (n+2*x)*P(n,x) = -(n^3-2*n^2-n+2+(2*n^3-n^2-8*n+7)*x+(n^3+4*n^2-11*n+6)*x^2+(3*n^2-2*n-1)*x^3+(2*n-2)*x^4)*P(n-2,x) + ((n^2-1)*x+(2*n^2+3*n-3)*x^2+(n^2+6*n-1)*x^3+(3*n+3)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2-n-1+(2*n^2+3*n-4)*x+(5*n-3)*x^2+2*x^3)*P(n-1,x) - (n*x+(n+2)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 2*n^2 - n + 2 + (2*n^3 - n^2 - 8*n + 7)*x + (n^3 + 4*n^2 - 11*n + 6)*x^2 + (3*n^2 - 2*n - 1)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + ((n^2 - 1)*x + (2*n^2 + 3*n - 3)*x^2 + (n^2 + 6*n - 1)*x^3 + (3*n + 3)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - n - 1 + (2*n^2 + 3*n - 4)*x + (5*n - 3)*x^2 + 2*x^3)*P[n - 1, x] - (n*x + (n + 2)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 2*n^2 - n + 2 + (2*n^3 - n^2 - 8*n + 7)*x + (n^3 + 4*n^2 - 11*n + 6)*x^2 + (3*n^2 - 2*n - 1)*x^3 + (2*n - 2)*x^4)*v[i - 2] + ((n^2 - 1)*x + (2*n^2 + 3*n - 3)*x^2 + (n^2 + 6*n - 1)*x^3 + (3*n + 3)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - n - 1 + (2*n^2 + 3*n - 4)*x + (5*n - 3)*x^2 + 2*x^3)*v[i - 1] - (n*x + (n + 2)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2*x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.