Real-rooted atlas

A143497 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 6\,x + 4\,x^{2} + 2\,n x + 5\,n x^{2} + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 4\,x^{3} - 2\,n x^{3}\right)\,P_{n-1}'(x) + \left(x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1 + 10\,x + 20\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01102031
111890120229
21282528408401961
314056033608400672019081
4154108010080453609072060480207775
517018902520017640063504010584006048002501801
61883080554405544003104640931392013305600665280032989969
7110847521108801496880119750405588352014370048017962560079833600472630861

All 15 rows

Technical details
Node id
n-82e4c94e68305e5a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a76d8ad579b60b88

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,  10,    20;
  1: 1,  18,    90,    120;
  2: 1,  28,   252,    840,     840;
  3: 1,  40,   560,   3360,    8400,      6720;
  4: 1,  54,  1080,  10080,   45360,     90720,     60480;
  5: 1,  70,  1890,  25200,  176400,    635040,   1058400,     604800;
  6: 1,  88,  3080,  55440,  554400,   3104640,   9313920,   13305600,     6652800;
  7: 1, 108,  4752, 110880, 1496880,  11975040,  55883520,  143700480,   179625600,    79833600;
  8: 1, 130,  7020, 205920, 3603600,  38918880, 259459200, 1037836800,  2335132800,  2594592000,  1037836800;
  9: 1, 154, 10010, 360360, 7927920, 110990880, 998917920, 5708102400, 19978358400, 39956716800, 39956716800, 14529715200;
%F Formula
The row polynomials satisfy P(n,x) = (1+(2*n+6)*x+(n^2+5*n+4)*x^2)*P(n-1,x) - (2*x^2+(2*n+4)*x^3)*d/dx P(n-1,x) + x^4*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1 + 10*x + 20*x^2.
%t Mathematica
P[0, x_] := 1 + 10*x + 20*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n + 6)*x + (n^2 + 5*n + 4)*x^2)*P[n - 1, x] - (2*x^2 + (2*n + 4)*x^3)*D[P[n - 1, x], x] + x^4*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 10*x + 20*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (2*n + 6)*x + (n^2 + 5*n + 4)*x^2)*v[i - 1] - (2*x^2 + (2*n + 4)*x^3)*deriv(v[i - 1], x) + x^4*deriv(deriv(v[i - 1], x), 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

Same fingerprint ?

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.