Real-rooted atlas

A136572 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(n + 6\,n x + 5\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1156
22205072
36904507501296
4244803600120001500031104
5120300030000150000375000375000933120
67202160027000018000006750000135000001125000033592320
750401764002646000220500001102500003307500005512500003937500001410877440
8403201612800282240002822400001764000000705600000017640000000252000000001575000000067722117120
936288016329600326592000381024000028576800000142884000000476280000000102060000000012757500000007087500000003656994324480
10362880018144000040824000005443200000047628000000028576800000001190700000000034020000000000637875000000007087500000000035437500000000219419659468800
11399168002195424000548856000008232840000008232840000000576298800000002881494000000001029105000000000257276250000000042879375000000004287937500000000194906250000000014481697524940800
1247900160028740096000790352640000131725440000001481911200000001185528960000000691558560000000029638224000000000926194500000000002058210000000000003087315000000000002806650000000000001169437500000000001042682221795737600
1362270208004047563520001214269056000022261599360000027826999200000002504429928000000016696199520000000083480997600000000031305374100000000008695937250000000000173918745000000000002371619250000000000019763493750000000000760134375000000000081329213300067532800
148717829120061024803840001983306124800003966612249600000545409184320000005454091843200000004090568882400000000233746793280000000001022642220600000000003408807402000000000008522018505000000000001549457910000000000000193682238750000000000014898633750000000000005320940625000000000006831653917205672755200
Technical details
Node id
n-b758fb00deafc303 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a8e18dd2f0709b33

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:      1,        5;
  2:      2,       20,        50;
  3:      6,       90,       450,        750;
  4:     24,      480,      3600,      12000,       15000;
  5:    120,     3000,     30000,     150000,      375000,       375000;
  6:    720,    21600,    270000,    1800000,     6750000,     13500000,     11250000;
  7:   5040,   176400,   2646000,   22050000,   110250000,    330750000,    551250000,     393750000;
  8:  40320,  1612800,  28224000,  282240000,  1764000000,   7056000000,  17640000000,   25200000000,   15750000000;
  9: 362880, 16329600, 326592000, 3810240000, 28576800000, 142884000000, 476280000000, 1020600000000, 1275750000000, 708750000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+6*n*x+5*n*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[((n + 6*n*x + 5*n*x^2)*P[n - 1, x])/(1 + 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] = ((n + 6*n*x + 5*n*x^2)*v[i - 1])/(1 + 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.