Real-rooted atlas

A367270 → basis A013611 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(10 + 8\,x - 26\,n - 20\,n x + 16\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(5 + 9\,x + 4\,x^{2} - 10\,n - 18\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
211819
310614448298
41011196812482564483
5972624480225608960128067006
6945262911203504002048005760061441004590
792683633727685003040385280016128003440642867215140980
8915373138411744676798086463744035929600115261441949696131072229419235
990941686432191808881633088100480665669656832029881958476726272106168325898243492894070
1090792282648190341601116798912014788945920122858265606606286848228286464048365568056033280262144053401180474
119101393676533576894401384513416002088759552002022687744001308334325765693140992016328294400291962880028835840011534336819367812748
1291551938286587491885728168721759008028565319552003159568051200238933662105612550532628484552025702401107836928001701314560014533263365033164812611254371022
1392365974135664390185325442027575638316838065588154880473488988544004099574233497625244717481984110883089940483426087075840719742566400962655682567197425664218103808194631201217324
1493426038382367030453897084824084790109452849656185164185668611293214464066935765049753647296152276172824415782096076891634587140096245297263411204509187178496531300876288351147130889395240963010887677683324

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-32 - 64\,x - 32\,x^{2} + 80\,n + 160\,n x + 80\,n x^{2} - 32\,n^{2} - 64\,n^{2} x - 32\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(22 + 16\,x - 48\,n - 32\,n x + 24\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-6e441e5fd2d6315b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-c12d924c8e8c7d11)
Fingerprint
f6-8f1a2f4c2ea52b05

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2:       11,         8;
  3:      106,       144,        48;
  4:     1011,      1968,      1248,        256;
  5:     9726,     24480,     22560,       8960,      1280;
  6:    94526,    291120,    350400,     204800,     57600,      6144;
  7:   926836,   3372768,   5003040,    3852800,   1612800,    344064,    28672;
  8:  9153731,  38411744,  67679808,   64637440,  35929600,  11526144,  1949696,   131072;
  9: 90941686, 432191808, 881633088, 1004806656, 696568320, 298819584, 76726272, 10616832, 589824;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (16*n^2-26*n+10+(12*n^2-20*n+8)*x)*P(n-1,x) - (10*n-5+(18*n-9)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 26*n + 10 + (12*n^2 - 20*n + 8)*x)*P[n - 1, x] - (10*n - 5 + (18*n - 9)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^2 - 26*n + 10 + (12*n^2 - 20*n + 8)*x)*v[i - 1] - (10*n - 5 + (18*n - 9)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023