Real-rooted atlas

A085881 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x + n\right)\,P_{n}(x) = \left(-4 - 4\,x - x^{2} + 6\,n + 7\,n x + 2\,n x^{2} + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
22418345
330036013515810
4504084005040126010519845
51058402268001890007560014175945612360
6266112069854407484400415800012474001871101039522733865
77783776024216192031783752022702680094594500227026802837835135135985134150

All 15 rows

Technical details
Node id
n-3ae1126e4bd3090b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           3,            1;
  2:          24,           18,            3;
  3:         300,          360,          135,           15;
  4:        5040,         8400,         5040,         1260,          105;
  5:      105840,       226800,       189000,        75600,        14175,          945;
  6:     2661120,      6985440,      7484400,      4158000,      1247400,       187110,       10395;
  7:    77837760,    242161920,    317837520,    227026800,     94594500,     22702680,     2837835,     135135;
  8:  2594592000,   9340531200,  14529715200,  12713500800,   6810804000,   2270268000,   454053600,   48648600,   2027025;
  9: 97037740800, 396972576000, 714550636800, 741015475200, 486291405600, 208410602400, 57891834000, 9924314400, 930404475, 34459425;
%F Formula
The row polynomials satisfy (n+1+x)*P(n,x) = (4*n^2+6*n-4+(2*n^2+7*n-4)*x+(2*n-1)*x^2)*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[((4*n^2 + 6*n - 4 + (2*n^2 + 7*n - 4)*x + (2*n - 1)*x^2)*P[n - 1, x])/(n + 1 + x)]];
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] = ((4*n^2 + 6*n - 4 + (2*n^2 + 7*n - 4)*x + (2*n - 1)*x^2)*v[i - 1])/(n + 1 + x)); 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.