Real-rooted atlas

A196347 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(n + 3\,n x + 3\,n x^{2} + n x^{3} + 3\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3} + n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
22161836
367216296336
424384129615366003840
51202400108001920015000432051840
6720172809720023040027000015552035280806400
75040141120952560282240044100003810240172872032256014192640
84032012902401016064036126720705600008128512055319040206438403265920278691840
936288013063680117573120487710720114307200016460236801493614080836075520264539520362880006038323200
1036288001451520001469664000696729600019051200000329204736003734035200027869184000132269760003628800000439084800143061811200
11399168001756339200197588160001053803520003293136000006638962176009036365184008430428160005334880320002195424000005312926080057480192003678732288000
1247900160022992076800284526950400168608563200059276448000001365729361920021687276441600242796331008001920556915200010538035200000382530677760082771476480080951270400102023508787200
136227020800323805081600437136860160028494847180800111307996800000288510327705600523592816947200683876332339200649148237337600445231987200000215492281804800699418976256001368076469760012204960768003035199386419200
1487178291200488198430720071399020492800507726367948800218163673728000062831138033664001282802401520640019148537305497600212055090863616001745309389824000010559121808435200456953731153920013407149403648002392172310528001961511552000096412215803904000
Technical details
Node id
n-d50b3afa8ba1682c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:      1,        4;
  2:      2,       16,        18;
  3:      6,       72,       162,        96;
  4:     24,      384,      1296,      1536,        600;
  5:    120,     2400,     10800,     19200,      15000,       4320;
  6:    720,    17280,     97200,    230400,     270000,     155520,      35280;
  7:   5040,   141120,    952560,   2822400,    4410000,    3810240,    1728720,    322560;
  8:  40320,  1290240,  10160640,  36126720,   70560000,   81285120,   55319040,  20643840,   3265920;
  9: 362880, 13063680, 117573120, 487710720, 1143072000, 1646023680, 1493614080, 836075520, 264539520, 36288000;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (n+(3*n^2+3*n)*x+(n^3+5*n^2+3*n)*x^2+(n^3+2*n^2+n)*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[((n + (3*n^2 + 3*n)*x + (n^3 + 5*n^2 + 3*n)*x^2 + (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*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] = ((n + (3*n^2 + 3*n)*x + (n^3 + 5*n^2 + 3*n)*x^2 + (n^3 + 2*n^2 + n)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*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.