Real-rooted atlas

A113413 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 8\,x + 5\,x^{2} + 10\,n x + 10\,n x^{2} - 2\,n x^{3} - 2\,n x^{4} + 8\,n^{2} x^{2} + 4\,n^{2} x^{3} - 2\,n^{2} x^{4} + 2\,n^{3} x^{3}\right)\,P_{n}(x) = \left(20 + 72\,x + 77\,x^{2} + 12\,x^{3} - 20\,x^{4} - 8\,x^{5} - 8\,n + 10\,n x + 80\,n x^{2} + 74\,n x^{3} + 6\,n x^{4} - 8\,n x^{5} - 12\,n^{2} x - 12\,n^{2} x^{2} + 24\,n^{2} x^{3} + 18\,n^{2} x^{4} - 4\,n^{3} x^{2} - 6\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(4 + 20\,x + 30\,x^{2} + 2\,x^{3} - 30\,x^{4} - 22\,x^{5} - 4\,x^{6} + 8\,n x + 36\,n x^{2} + 48\,n x^{3} + 10\,n x^{4} - 14\,n x^{5} - 4\,n x^{6} + 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 18\,n^{2} x^{4} + 4\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
27241243
3179210832249
441304576384801385
5999322448262412001927495
6239272891801369610000345644839747
757777403178861024620003398494081024207545

All 15 rows

Technical details
Node id
n-4c7c80526f3eab02 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8f90520aa939060a

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,     4;
  2:    7,    24,     12;
  3:   17,    92,    108,     32;
  4:   41,   304,    576,    384,      80;
  5:   99,   932,   2448,   2624,    1200,     192;
  6:  239,  2728,   9180,  13696,   10000,    3456,    448;
  7:  577,  7740,  31788,  61024,   62000,   33984,   9408,   1024;
  8: 1393, 21472, 104160, 244992,  320800,  244224, 106624,  24576,  2304;
  9: 3363, 58564, 327744, 913536, 1468640, 1441152, 871808, 315392, 62208, 5120;
%F Formula
The row polynomials satisfy (4+(10*n+8)*x+(8*n^2+10*n+5)*x^2+(2*n^3+4*n^2-2*n)*x^3-(2*n^2+2*n)*x^4)*P(n,x) = -(8*n-20+(12*n^2-10*n-72)*x+(4*n^3+12*n^2-80*n-77)*x^2+(6*n^3-24*n^2-74*n-12)*x^3-(18*n^2+6*n-20)*x^4+(8*n+8)*x^5)*P(n-1,x) + (4+(8*n+20)*x+(4*n^2+36*n+30)*x^2+(16*n^2+48*n+2)*x^3+(18*n^2+10*n-30)*x^4+(4*n^2-14*n-22)*x^5-(4*n+4)*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[(-(8*n - 20 + (12*n^2 - 10*n - 72)*x + (4*n^3 + 12*n^2 - 80*n - 77)*x^2 + (6*n^3 - 24*n^2 - 74*n - 12)*x^3 - (18*n^2 + 6*n - 20)*x^4 + (8*n + 8)*x^5)*P[n - 1, x] + (4 + (8*n + 20)*x + (4*n^2 + 36*n + 30)*x^2 + (16*n^2 + 48*n + 2)*x^3 + (18*n^2 + 10*n - 30)*x^4 + (4*n^2 - 14*n - 22)*x^5 - (4*n + 4)*x^6)*D[P[n - 1, x], x])/(4 + (10*n + 8)*x + (8*n^2 + 10*n + 5)*x^2 + (2*n^3 + 4*n^2 - 2*n)*x^3 - (2*n^2 + 2*n)*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] = (-(8*n - 20 + (12*n^2 - 10*n - 72)*x + (4*n^3 + 12*n^2 - 80*n - 77)*x^2 + (6*n^3 - 24*n^2 - 74*n - 12)*x^3 - (18*n^2 + 6*n - 20)*x^4 + (8*n + 8)*x^5)*v[i - 1] + (4 + (8*n + 20)*x + (4*n^2 + 36*n + 30)*x^2 + (16*n^2 + 48*n + 2)*x^3 + (18*n^2 + 10*n - 30)*x^4 + (4*n^2 - 14*n - 22)*x^5 - (4*n + 4)*x^6)*deriv(v[i - 1], x))/(4 + (10*n + 8)*x + (8*n^2 + 10*n + 5)*x^2 + (2*n^3 + 4*n^2 - 2*n)*x^3 - (2*n^2 + 2*n)*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.