Real-rooted atlas

A136124 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 8\,x + 6\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13710
2125256120
3603687485041680
43602808816810504504026880
52520236168806416337615082455440483840
6201602191689879042363776316710422534086652809676800
7181440223459211740320341167365923302461453024352818248648640212889600
81814400248860801486874885055318401069964544144379257612131605765804319361210809605109350400
9199584003009628802008853568779116256019352587584319311864963500165497624582159936100383696001816214400132843110400
10239500800393088896028921648896125635607808356889651968692815639296930918843648855012398848513774994176182408486400290594304003719607091200
113113510400551730700804428065479682124930932736677537080832015073727394304238798108354562694078152704021214285704704111054852241923478716864000494010316800111588212736000
12435891456008284661683207192406932992377195509217281331039481057283329888580700166056444447938568070075364167687819340673525765373339358233602486007701314566953304798720088921857024003570822807552000
136538371840001325518629120012362696119296070244899555276827132306890987527523807885059072154114549752381442361235749733478427062154202207232229167563170493441393835633261568057664103523056641454510697292800168951528345600121407975456768000
1410461394944000225159724339200224313510491136013711723152703488574606709366353921746455399429201923970594373049917446860268194621194249052416171820113929079111851169341446813488276628377603710296922892042241386003782027980803179343839938560033790305669120004370687116443648000
Technical details
Node id
n-4d17ebe7f25516c7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9286d4fca5ff723a

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:        3,         7;
  2:       12,        52,         56;
  3:       60,       368,        748,        504;
  4:      360,      2808,       8168,      10504,        5040;
  5:     2520,     23616,      88064,     163376,      150824,       55440;
  6:    20160,    219168,     987904,    2363776,     3167104,     2253408,      665280;
  7:   181440,   2234592,   11740320,   34116736,    59233024,    61453024,    35281824,     8648640;
  8:  1814400,  24886080,  148687488,  505531840,  1069964544,  1443792576,  1213160576,   580431936,   121080960;
  9: 19958400, 300962880, 2008853568, 7791162560, 19352587584, 31931186496, 35001654976, 24582159936, 10038369600, 1816214400;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+2+(2*n+8)*x+(n+6)*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 + 2 + (2*n + 8)*x + (n + 6)*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 + 2 + (2*n + 8)*x + (n + 6)*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.