Real-rooted atlas

A143491 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 12\,x + 8\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 12 + 52\,x + 56\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0125256120
1603687485041680
23602808816810504504026880
32520236168806416337615082455440483840
4201602191689879042363776316710422534086652809676800
5181440223459211740320341167365923302461453024352818248648640212889600
61814400248860801486874885055318401069964544144379257612131605765804319361210809605109350400
7199584003009628802008853568779116256019352587584319311864963500165497624582159936100383696001816214400132843110400

All 15 rows

Technical details
Node id
n-820004861ef354c6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9286d4fca5ff723a

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         12,          52,           56;
  1:         60,         368,          748,           504;
  2:        360,        2808,         8168,         10504,          5040;
  3:       2520,       23616,        88064,        163376,        150824,          55440;
  4:      20160,      219168,       987904,       2363776,       3167104,        2253408,         665280;
  5:     181440,     2234592,     11740320,      34116736,      59233024,       61453024,       35281824,        8648640;
  6:    1814400,    24886080,    148687488,     505531840,    1069964544,     1443792576,     1213160576,      580431936,      121080960;
  7:   19958400,   300962880,   2008853568,    7791162560,   19352587584,    31931186496,    35001654976,    24582159936,    10038369600,     1816214400;
  8:  239500800,  3930888960,  28921648896,  125635607808,  356889651968,   692815639296,   930918843648,   855012398848,   513774994176,   182408486400,   29059430400;
  9: 3113510400, 55173070080, 442806547968, 2124930932736, 6775370808320, 15073727394304, 23879810835456, 26940781527040, 21214285704704, 11105485224192, 3478716864000, 494010316800;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+4+(2*n+12)*x+(n+8)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 52*x + 56*x^2.
%t Mathematica
P[0, x_] := 12 + 52*x + 56*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (2*n + 12)*x + (n + 8)*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] = 12 + 52*x + 56*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (2*n + 12)*x + (n + 8)*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.