Real-rooted atlas

A196347 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1257
28405098
3483609007502058
4384384014400240001500057624
53840480002400006000007500003750002016840
64608069120043200001440000027000000270000001125000084707280
76451201128960084672000352800000882000000132300000011025000003937500004150656720
810321920206438400180633600090316800002822400000056448000000705600000005040000000015750000000232436776320
9185794560418037760041803776000243855360000914457600000228614400000038102400000004082400000000255150000000070875000000014643516908160
10371589120092897280000104509440000069672960000003048192000000091445760000000190512000000000272160000000000255150000000000141750000000000354375000000001025046183571200
11817496064002248114176000281014272000002107607040000001053803520000000368831232000000092207808000000001646568000000000020582100000000000171517500000000008575875000000000194906250000000078928556134982400
121961990553600588597166080008093211033600006744342528000000379369267200000001517477068800000004425974784000000009484231680000000001481911200000000000164656800000000000012349260000000000005613300000000000001169437500000000006629998715338521600
135101175439360016578820177920002486823026688000022795877744640000014247423590400000006411340615680000000213711353856000000005342783846400000000010017719712000000000013913499600000000000013913499600000000000094864770000000000000395269875000000000007601343750000000000603329883095805465600
141428329123020800499915193057280008123621887180800008123621887180800000558499004743680000002792495023718400000001047185633894400000000299195895398400000000065449102118400000000001090818368640000000000013635229608000000000000123956632800000000000007747289550000000000000297972675000000000000053209406250000000000059126328543388935628800
Technical details
Node id
n-44803ce4a1b16505 · 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:         2,          5;
  2:         8,         40,          50;
  3:        48,        360,         900,          750;
  4:       384,       3840,       14400,        24000,        15000;
  5:      3840,      48000,      240000,       600000,       750000,        375000;
  6:     46080,     691200,     4320000,     14400000,     27000000,      27000000,      11250000;
  7:    645120,   11289600,    84672000,    352800000,    882000000,    1323000000,    1102500000,     393750000;
  8:  10321920,  206438400,  1806336000,   9031680000,  28224000000,   56448000000,   70560000000,   50400000000,   15750000000;
  9: 185794560, 4180377600, 41803776000, 243855360000, 914457600000, 2286144000000, 3810240000000, 4082400000000, 2551500000000, 708750000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+7*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[((2*n + 7*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] = ((2*n + 7*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.