Real-rooted atlas

A178821 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - 2\,n x + 6\,n^{2} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(8 + 48\,x + 96\,x^{2} + 64\,x^{3} + 2\,n + 60\,n x + 184\,n x^{2} + 144\,n x^{3} + 12\,n^{2} x + 72\,n^{2} x^{2} + 96\,n^{2} x^{3} + 8\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1108090
26096021603180
32806720302403584073080
41120358402419205734404480001300320
54032161280145152051609608064000464486419486656
613440645120725760034406400806400009289728042147840258007680
74224023654403193344018923520059136000010218700809272524803460300803110088960

All 15 rows

Technical details
Node id
n-48dac2e4a8f752f8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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:     10,       80;
  2:     60,      960,      2160;
  3:    280,     6720,     30240,      35840;
  4:   1120,    35840,    241920,     573440,      448000;
  5:   4032,   161280,   1451520,    5160960,     8064000,     4644864;
  6:  13440,   645120,   7257600,   34406400,    80640000,    92897280,    42147840;
  7:  42240,  2365440,  31933440,  189235200,   591360000,  1021870080,   927252480,    346030080;
  8: 126720,  8110080, 127733760,  908328960,  3548160000,  8174960640, 11127029760,   8304721920,  2627665920;
  9: 366080, 26357760, 474439680, 3936092160, 18450432000, 53137244160, 96434257920, 107961384960, 68319313920, 18743296000;
%F Formula
The row polynomials satisfy (n+(6*n^2-2*n)*x+4*n^3*x^2)*P(n,x) = (2*n+8+(12*n^2+60*n+48)*x+(8*n^3+72*n^2+184*n+96)*x^2+(16*n^3+96*n^2+144*n+64)*x^3)*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[((2*n + 8 + (12*n^2 + 60*n + 48)*x + (8*n^3 + 72*n^2 + 184*n + 96)*x^2 + (16*n^3 + 96*n^2 + 144*n + 64)*x^3)*P[n - 1, x])/(n + (6*n^2 - 2*n)*x + 4*n^3*x^2)]];
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] = ((2*n + 8 + (12*n^2 + 60*n + 48)*x + (8*n^3 + 72*n^2 + 184*n + 96)*x^2 + (16*n^3 + 96*n^2 + 144*n + 64)*x^3)*v[i - 1])/(n + (6*n^2 - 2*n)*x + 4*n^3*x^2)); 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

Same fingerprint ?

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.