Real-rooted atlas

A081578 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n x + 6\,n x^{2} + 2\,n x^{3} + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(8 + 39\,x + 62\,x^{2} + 36\,x^{3} + 8\,x^{4} - 8\,n - 21\,n x + n x^{2} + 29\,n x^{3} + 12\,n x^{4} - 12\,n^{2} x - 29\,n^{2} x^{2} - 10\,n^{2} x^{3} + 4\,n^{2} x^{4} - 4\,n^{3} x^{2} - 7\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(4 + 28\,x + 73\,x^{2} + 85\,x^{3} + 40\,x^{4} + 4\,x^{5} + 8\,n x + 44\,n x^{2} + 82\,n x^{3} + 55\,n x^{4} + 6\,n x^{5} + 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 17\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
27281247
32012014432316
461488936544802109
5182182048484832176019213634
65476564223563171219920518444886731
71640229609532817632016064071424143361024543672

All 15 rows

Technical details
Node id
n-439d83864d3a6aad · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-510c514856bfe6b0

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:     2,      4;
  2:     7,     28,      12;
  3:    20,    120,     144,      32;
  4:    61,    488,     936,     544,      80;
  5:   182,   1820,    4848,    4832,    1760,     192;
  6:   547,   6564,   22356,   31712,   19920,    5184,     448;
  7:  1640,  22960,   95328,  176320,  160640,   71424,   14336,   1024;
  8:  4921,  78736,  385680,  880832, 1059680,  687360,  232960,  37888,  2304;
  9: 14762, 265716, 1500192, 4081728, 6125760, 5288832, 2612736, 709632, 96768, 5120;
%F Formula
The row polynomials satisfy (3*n*x+(3*n^2+6*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(8*n-8+(12*n^2+21*n-39)*x+(4*n^3+29*n^2-n-62)*x^2+(7*n^3+10*n^2-29*n-36)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (4+(8*n+28)*x+(4*n^2+44*n+73)*x^2+(16*n^2+82*n+85)*x^3+(17*n^2+55*n+40)*x^4+(2*n^2+6*n+4)*x^5)*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 - 8 + (12*n^2 + 21*n - 39)*x + (4*n^3 + 29*n^2 - n - 62)*x^2 + (7*n^3 + 10*n^2 - 29*n - 36)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (4 + (8*n + 28)*x + (4*n^2 + 44*n + 73)*x^2 + (16*n^2 + 82*n + 85)*x^3 + (17*n^2 + 55*n + 40)*x^4 + (2*n^2 + 6*n + 4)*x^5)*D[P[n - 1, x], x])/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];
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 - 8 + (12*n^2 + 21*n - 39)*x + (4*n^3 + 29*n^2 - n - 62)*x^2 + (7*n^3 + 10*n^2 - 29*n - 36)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (4 + (8*n + 28)*x + (4*n^2 + 44*n + 73)*x^2 + (16*n^2 + 82*n + 85)*x^3 + (17*n^2 + 55*n + 40)*x^4 + (2*n^2 + 6*n + 4)*x^5)*deriv(v[i - 1], x))/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)); 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.