Real-rooted atlas

A178822 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\,n + 3\,n^{2} x\right)\,P_{n}(x) = \left(20 + 60\,x + 45\,x^{2} + 4\,n + 42\,n x + 54\,n x^{2} + 6\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123648
2845045671155
344840329072604819600
42016241928164810886451030267750
58064120960544320108864010206003674163150000
62956853222429937607983360112266008083152235758633206250
71013762128896143700484790016089812800969978245658206413856832321750000
8329472790732862270208249080832583783200840647808735566832360277632759960632915859375
910250242767564824908083211623772163269185920588453465668652904325043886848212788976439405366025025000000
1030750729225216093405312049816166401634592960035307207936514896782405043886848031918346460118216098001950565617205283203125
118945664295206912332107776019926466560747242496001883051089923295339407364035109478403404623622401891457568006241809974492853371521620937500000
12253460489124577281129166438475277762560317578060800914624815104186735899750427438744453122893930079040214365191040010611076956483157014631684275123980412388085937500
1370189056273737318436954537984270999945216127031224320041158116679689603560558592164632466718722083629656908819292867193600127332923477765682626337024153904463294419124223249692039062500000
141905131528001552384117022703616936181628928482718652416017377871486976456169126533128937191050444813196321160422414662579067136012096627730387271979933602304292418480259367267204834848834245452980667133789062500
Technical details
Node id
n-d93f88bb6eca3e5b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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:      12,       36;
  2:      84,      504,       567;
  3:     448,     4032,      9072,       6048;
  4:    2016,    24192,     81648,     108864,      51030;
  5:    8064,   120960,    544320,    1088640,    1020600,     367416;
  6:   29568,   532224,   2993760,    7983360,   11226600,    8083152,    2357586;
  7:  101376,  2128896,  14370048,   47900160,   89812800,   96997824,   56582064,   13856832;
  8:  329472,  7907328,  62270208,  249080832,  583783200,  840647808,  735566832,  360277632,   75996063;
  9: 1025024, 27675648, 249080832, 1162377216, 3269185920, 5884534656, 6865290432, 5043886848, 2127889764, 394053660;
%F Formula
The row polynomials satisfy (2*n+3*n^2*x)*P(n,x) = (4*n+20+(6*n^2+42*n+60)*x+(9*n^2+54*n+45)*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[((4*n + 20 + (6*n^2 + 42*n + 60)*x + (9*n^2 + 54*n + 45)*x^2)*P[n - 1, x])/(2*n + 3*n^2*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] = ((4*n + 20 + (6*n^2 + 42*n + 60)*x + (9*n^2 + 54*n + 45)*x^2)*v[i - 1])/(2*n + 3*n^2*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.