Real-rooted atlas

A178822 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + n x\right)\,P_{n}(x) = \left(10 + 35\,x + 25\,x^{2} + 2\,n + 7\,n x + 5\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123042
2844205251029
344833608400700019208
42016201607560012600078750302526
58064100800504000126000015750007875004235364
629568443520277200092400001732500017325000721875054353838
71013761774080133056005544000013860000020790000017325000061875000652246056
83294726589440576576002882880009009000001801800000225225000016087500005027343757419298887
91025024230630402306304001345344000504504000012612600000210210000002252250000014076562500391015625080787921214
103075072768768008648640005765760000252252000007567560000015765750000022522500000021114843750011730468750029326171875848273172747
11894566424600576030750720002306304000011531520000040360320000010090080000001801800000000225225000000018768750000009384375000002132812500008636963213424
1225346048760381440104552448008712704000049008960000019603584000005717712000000122522400000001914412500000021271250000000159534375000007251562500000151074218750085649885199788
13701890562281144320342171648003136573440001960358400000882161280000029405376000000735134400000001378377000000001914412500000001914412500000001305281250000005438671875000010458984375000830145041167176
14190513152666796032010835435520010835435520007449361920000372468096000001396755360000003990729600000008729721000000001454953500000000181869187500000016533562500000001033347656250000397441406250000709716796875007886377891088172
Technical details
Node id
n-21ae99bfbb1128ab · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9629d1e9c04dbfbe

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,       30;
  2:      84,      420,       525;
  3:     448,     3360,      8400,       7000;
  4:    2016,    20160,     75600,     126000,      78750;
  5:    8064,   100800,    504000,    1260000,    1575000,      787500;
  6:   29568,   443520,   2772000,    9240000,   17325000,    17325000,     7218750;
  7:  101376,  1774080,  13305600,   55440000,  138600000,   207900000,   173250000,    61875000;
  8:  329472,  6589440,  57657600,  288288000,  900900000,  1801800000,  2252250000,  1608750000,   502734375;
  9: 1025024, 23063040, 230630400, 1345344000, 5045040000, 12612600000, 21021000000, 22522500000, 14076562500, 3910156250;
%F Formula
The row polynomials satisfy (n+n*x)*P(n,x) = (2*n+10+(7*n+35)*x+(5*n+25)*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[((2*n + 10 + (7*n + 35)*x + (5*n + 25)*x^2)*P[n - 1, x])/(n + n*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] = ((2*n + 10 + (7*n + 35)*x + (5*n + 25)*x^2)*v[i - 1])/(n + n*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.