Real-rooted atlas

A178821 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n + 3\,n x + 3\,n^{2}\right)\,P_{n}(x) = \left(16 + 48\,x + 36\,x^{2} + 28\,n + 48\,n x + 9\,n x^{2} + 6\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1251540
2240450135825
315405040472594512250
478403696060480378005670148750
534272211680498960544320255150306181575000
61344001028160317520049896004082400153090015309015093750
74857604435200169646403492720041164200269438408419950721710134062500

All 15 rows

Technical details
Node id
n-7ecf7fa89e3ab64f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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:      25,       15;
  2:     240,      450,       135;
  3:    1540,     5040,      4725,        945;
  4:    7840,    36960,     60480,      37800,       5670;
  5:   34272,   211680,    498960,     544320,     255150,      30618;
  6:  134400,  1028160,   3175200,    4989600,    4082400,    1530900,     153090;
  7:  485760,  4435200,  16964640,   34927200,   41164200,   26943840,    8419950,    721710;
  8: 1647360, 17487360,  79833600,  203575680,  314344800,  296382240,  161663040,  43302600,   3247695;
  9: 5308160, 64247040, 341003520, 1037836800, 1984862880, 2451889440, 1926484560, 900694080, 211100175, 14073345;
%F Formula
The row polynomials satisfy (3*n^2-n+3*n*x)*P(n,x) = (6*n^2+28*n+16+(9*n^2+48*n+48)*x+(9*n+36)*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[((6*n^2 + 28*n + 16 + (9*n^2 + 48*n + 48)*x + (9*n + 36)*x^2)*P[n - 1, x])/(3*n^2 - n + 3*n*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] = ((6*n^2 + 28*n + 16 + (9*n^2 + 48*n + 48)*x + (9*n + 36)*x^2)*v[i - 1])/(3*n^2 - n + 3*n*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.