Real-rooted atlas

A196347 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2243660
32166484861350
423041036815552777636000
5288001728003888003888001458001125000
6414720311040093312001399680010497600314928040500000
7677376060963840228614400457228800514382400308629440771573601653750000
8123863040130056192058525286401463132160021946982400197522841609876142080211631616075600000000
92508226560300987187201580182732804740548198408888527872001066623344640799967508480342843217920642831033603827250000000
105573836800075246796800045148078080001580182732800035554111488000533311672320005333116723200034284321792000128566206720002142770112000212625000000000
11134886850560020233027584000136572936192000546291744768000143401583001600025812284940288003226535617536000276560195788800015556511013120005185503671040007778255506560012863812500000000
12353158299648005827111944192004370333958144000196665028116480005899950843494400012389896771338240018584845157007360019912334096793600014934250572595200074671252862976000224013758588928003054733071667200841995000000000000
13994729210675200179051257921536001477172877852672007385864389263360002492729231376384000598255015530332160010469462771780812800134607378494324736001261944173384294400084129611558952960003785832520152883200103249977822351360012906247227793920059290481250000000000
14299949115834368005849007758770176005264106982893158400289525884059123712001085722065221713920002931449576098627584005862899152197255168008794348728295882752009893642319332868096008244701932777390080004946821159666434048002023699565318086656005059248913295216640058375948999560192004469590125000000000000
Technical details
Node id
n-69e4e6e1680510e7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f06e51c139cfae05

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:          0;
  1:          3;
  2:         24,          36;
  3:        216,         648,          486;
  4:       2304,       10368,        15552,         7776;
  5:      28800,      172800,       388800,       388800,       145800;
  6:     414720,     3110400,      9331200,     13996800,     10497600,       3149280;
  7:    6773760,    60963840,    228614400,    457228800,    514382400,     308629440,     77157360;
  8:  123863040,  1300561920,   5852528640,  14631321600,  21946982400,   19752284160,   9876142080,   2116316160;
  9: 2508226560, 30098718720, 158018273280, 474054819840, 888852787200, 1066623344640, 799967508480, 342843217920, 64283103360;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n^2+3*n^2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n^2*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 3*n^2*x)*v[i - 1])/(n - 1)); 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.