Real-rooted atlas

A035607 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-32 - 72\,x - 46\,x^{2} - 4\,x^{3} + 2\,x^{4} + 8\,n x - 4\,n x^{2} - 10\,n x^{3} - 3\,n x^{4} + 8\,n^{2} x^{2} + 2\,n^{2} x^{3} + n^{2} x^{4} + 2\,n^{3} x^{3}\right)\,P_{n}(x) = \left(32 + 72\,x + 46\,x^{2} + 4\,x^{3} - 2\,x^{4} - 64\,n - 88\,n x - 24\,n x^{2} + 6\,n x^{3} + 3\,n x^{4} - 48\,n^{2} x - 56\,n^{2} x^{2} - 14\,n^{2} x^{3} - n^{2} x^{4} - 8\,n^{3} x^{2} - 6\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(64 + 224\,x + 296\,x^{2} + 180\,x^{3} + 48\,x^{4} + 4\,x^{5} + 64\,n x + 176\,n x^{2} + 176\,n x^{3} + 78\,n x^{4} + 15\,n x^{5} + n x^{6} + 16\,n^{2} x^{2} + 32\,n^{2} x^{3} + 18\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2347
3716629
41756428123
5411761988010505
699524786488130122039
7239150428202368980192148117

All 15 rows

Technical details
Node id
n-ad67261ee2d3133a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c090123fef931ef5

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    1;
  2:    3,     4;
  3:    7,    16,     6;
  4:   17,    56,    42,     8;
  5:   41,   176,   198,    80,    10;
  6:   99,   524,   786,   488,   130,    12;
  7:  239,  1504,  2820,  2368,   980,   192,   14;
  8:  577,  4208,  9468, 10032,  5660,  1728,  266,  16;
  9: 1393, 11552, 30324, 38816, 27800, 11616, 2786, 352, 18;
%F Formula
The row polynomials satisfy -(32-(8*n-72)*x-(8*n^2-4*n-46)*x^2-(2*n^3+2*n^2-10*n-4)*x^3-(n^2-3*n+2)*x^4)*P(n,x) = -(64*n-32+(48*n^2+88*n-72)*x+(8*n^3+56*n^2+24*n-46)*x^2+(6*n^3+14*n^2-6*n-4)*x^3+(n^2-3*n+2)*x^4)*P(n-1,x) + (64+(64*n+224)*x+(16*n^2+176*n+296)*x^2+(32*n^2+176*n+180)*x^3+(18*n^2+78*n+48)*x^4+(2*n^2+15*n+4)*x^5+n*x^6)*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[(-(64*n - 32 + (48*n^2 + 88*n - 72)*x + (8*n^3 + 56*n^2 + 24*n - 46)*x^2 + (6*n^3 + 14*n^2 - 6*n - 4)*x^3 + (n^2 - 3*n + 2)*x^4)*P[n - 1, x] + (64 + (64*n + 224)*x + (16*n^2 + 176*n + 296)*x^2 + (32*n^2 + 176*n + 180)*x^3 + (18*n^2 + 78*n + 48)*x^4 + (2*n^2 + 15*n + 4)*x^5 + n*x^6)*D[P[n - 1, x], x])/(-32 + (8*n - 72)*x + (8*n^2 - 4*n - 46)*x^2 + (2*n^3 + 2*n^2 - 10*n - 4)*x^3 + (n^2 - 3*n + 2)*x^4)]];
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] = (-(64*n - 32 + (48*n^2 + 88*n - 72)*x + (8*n^3 + 56*n^2 + 24*n - 46)*x^2 + (6*n^3 + 14*n^2 - 6*n - 4)*x^3 + (n^2 - 3*n + 2)*x^4)*v[i - 1] + (64 + (64*n + 224)*x + (16*n^2 + 176*n + 296)*x^2 + (32*n^2 + 176*n + 180)*x^3 + (18*n^2 + 78*n + 48)*x^4 + (2*n^2 + 15*n + 4)*x^5 + n*x^6)*deriv(v[i - 1], x))/(-32 + (8*n - 72)*x + (8*n^2 - 4*n - 46)*x^2 + (2*n^3 + 2*n^2 - 10*n - 4)*x^3 + (n^2 - 3*n + 2)*x^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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.