Real-rooted atlas

A059364 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 6\,n x - 9\,n x^{2} + 4\,n^{2} + 18\,n^{2} x + 18\,n^{2} x^{2} - 4\,n^{3} - 12\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 6\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2333669
33668464861698
445541639819602777648330
5642693169985845774775761458001589220
610226076434658161584472023833612640860314928059644188
718176940139469364445068432756011556720869220365788872771573602522541744
8357656580324347230812587060448270943862763493512342026979444360115537635362116316160118867223088
9772587949580939429172370514776986967927515372157820288179516445074416481069374004524396726139008642831033606180201171360
101818853416452163280782312114230846294463514706890225269439542344145913499600192288001461455268444995500502848147394523620802142770112000351597159548640
114637090106810617598905417463698133452017021310998308447420304692606613323048508549572703865357309595452010405259054612387220094653604960725897302991856007778255506560021731868014024448
1212731168182635018778677512339701258036771929683450525998280867196135168075533464650252898579988715186337668648091656462321729787008836640214366551130418616951218718994828802529788372438976030547330716672001450417675881855744
1337457436989558256063274046577249044952519186820176320183810749868496606112680455595821355131541275946614556422062393888796126543723737078278466705872199037912855340102121185750775693751112047637711650463390721158787755027233280129062472277939200103973030402528060928
141175918052301476752072991944219691510168560932135215179978370646191923793886828323273964908842644568954879520721160991012426464798892880241311678324336697675211264169878689593803814860012727113182761082990406859662221568568468322518721782760947553285646765454979935104058375948999560192007968147354422325987840
Technical details
Node id
n-967d4341ec1a993c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a733e5459162abd5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          0;
  1:          3;
  2:         33,          36;
  3:        366,         846,          486;
  4:       4554,       16398,        19602,         7776;
  5:      64269,      316998,       584577,       477576,        145800;
  6:    1022607,     6434658,     16158447,     20238336,      12640860,       3149280;
  7:   18176940,   139469364,    445068432,    756011556,     720869220,     365788872,      77157360;
  8:  357656580,  3243472308,  12587060448,  27094386276,   34935123420,   26979444360,   11553763536,   2116316160;
  9: 7725879495, 80939429172, 370514776986, 967927515372, 1578202881795, 1644507441648, 1069374004524, 396726139008, 64283103360;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(12*n^3-18*n^2+6*n)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(6*n^2-6*n)*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[(-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*n)*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] = (-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*n)*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

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.