Real-rooted atlas

A225471 → basis A008277 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20 + 10\,x - x^{2} + 36\,n - 8\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(5\,x - x^{2} - 4\,n x\right)\,P_{n-2}'(x) + \left(-5 + 2\,x + 8\,n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
23226361
3384402814871
4614476022061184515996
51228801719305674569803506358891
62949120454401017332652641201852559479509641
7825753601376745305881396510498740922950418329318290528316

All 15 rows

Technical details
Node id
n-7c42936e45fe9737 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e06e9d2c9cbd3172

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:          32,           26,           3;
  3:         384,          402,          81,           4;
  4:        6144,         7602,        2061,         184,          5;
  5:      122880,       171930,       56745,        6980,        350,         6;
  6:     2949120,      4544010,     1733265,      264120,      18525,       594,       7;
  7:    82575360,    137674530,    58813965,    10498740,     922950,     41832,     931,      8;
  8:  2642411520,   4708410210,  2206186605,   445874520,   46349625,   2655324,   84182,   1376,    9;
  9: 95126814720, 179496000810, 90897706305, 20330688420, 2411328150, 164487834, 6633522, 155376, 1944, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^2-36*n+20+(8*n-10)*x+x^2)*P(n-2,x) - ((4*n-5)*x+x^2)*d/dx P(n-2,x) + (8*n-5+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 36*n + 20 + (8*n - 10)*x + x^2)*P[n - 2, x] - ((4*n - 5)*x + x^2)*D[P[n - 2, x], x] + (8*n - 5 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], 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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 36*n + 20 + (8*n - 10)*x + x^2)*v[i - 2] - ((4*n - 5)*x + x^2)*deriv(v[i - 2], x) + (8*n - 5 + 2*x)*v[i - 1] + x*deriv(v[i - 1], 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.