Real-rooted atlas

A094638 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 8\,n - 5\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(n - 6\,n x + 12\,n x^{2} - 8\,n x^{3} - 2\,n^{2} + 15\,n^{2} x - 36\,n^{2} x^{2} + 28\,n^{2} x^{3} + n^{3} - 12\,n^{3} x + 39\,n^{3} x^{2} - 38\,n^{3} x^{3} + 3\,n^{4} x - 18\,n^{4} x^{2} + 25\,n^{4} x^{3} + 3\,n^{5} x^{2} - 8\,n^{5} x^{3} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(-7\,n + 18\,n x - 12\,n x^{2} + 9\,n^{2} - 39\,n^{2} x + 36\,n^{2} x^{2} - 3\,n^{3} + 27\,n^{3} x - 39\,n^{3} x^{2} - 6\,n^{4} x + 18\,n^{4} x^{2} - 3\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-4 + 14\,n - 12\,n x - 12\,n^{2} + 24\,n^{2} x + 3\,n^{3} - 15\,n^{3} x + 3\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 7 + 18\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2718631
3137810524220
4212307056801201756
53154029556380497072015596
643109293453542061040408245040152804
757198824486143220427805627648373716403201639240

All 15 rows

Technical details
Node id
n-64b3e0a8d9bdbfc1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d4e0a6d76fe35fcf

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:  7,   18,      6;
  3: 13,   78,    105,      24;
  4: 21,  230,    705,     680,     120;
  5: 31,  540,   2955,    6380,    4970,      720;
  6: 43, 1092,   9345,   35420,   61040,    40824,     5040;
  7: 57, 1988,  24486,  143220,  427805,   627648,   373716,    40320;
  8: 73, 3348,  56070,  467124, 2130765,  5364912,  6963572,  3779712,   362880;
  9: 91, 5310, 116010, 1303260, 8396115, 32010510, 70736960, 83295072, 41898384, 3628800;
%F Formula
The row polynomials satisfy (n^3-5*n^2+8*n-4)*P(n,x) = (n^3-2*n^2+n+(3*n^4-12*n^3+15*n^2-6*n)*x+(3*n^5-18*n^4+39*n^3-36*n^2+12*n)*x^2+(n^6-8*n^5+25*n^4-38*n^3+28*n^2-8*n)*x^3)*P(n-3,x) - (3*n^3-9*n^2+7*n+(6*n^4-27*n^3+39*n^2-18*n)*x+(3*n^5-18*n^4+39*n^3-36*n^2+12*n)*x^2)*P(n-2,x) + (3*n^3-12*n^2+14*n-4+(3*n^4-15*n^3+24*n^2-12*n)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 7 + 18*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 7 + 18*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 2*n^2 + n + (3*n^4 - 12*n^3 + 15*n^2 - 6*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2 + (n^6 - 8*n^5 + 25*n^4 - 38*n^3 + 28*n^2 - 8*n)*x^3)*P[n - 3, x] - (3*n^3 - 9*n^2 + 7*n + (6*n^4 - 27*n^3 + 39*n^2 - 18*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2)*P[n - 2, x] + (3*n^3 - 12*n^2 + 14*n - 4 + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x)*P[n - 1, x])/(n^3 - 5*n^2 + 8*n - 4)]];
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] = 3 + 2*x; v[3] = 7 + 18*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 2*n^2 + n + (3*n^4 - 12*n^3 + 15*n^2 - 6*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2 + (n^6 - 8*n^5 + 25*n^4 - 38*n^3 + 28*n^2 - 8*n)*x^3)*v[i - 3] - (3*n^3 - 9*n^2 + 7*n + (6*n^4 - 27*n^3 + 39*n^2 - 18*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2)*v[i - 2] + (3*n^3 - 12*n^2 + 14*n - 4 + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x)*v[i - 1])/(n^3 - 5*n^2 + 8*n - 4)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.