Real-rooted atlas

A143491 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 11\,x + 7\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 12 + 45\,x + 42\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
012454299
1603215703361287
2360246663097146302419305
3252020862688231131129262830240328185
4201601946167800661661949198525612608283326406235515
518144019934649355986243183333781069235170524181236963991680130945815
6181440022293360119474892364811148694245249843244236638453772275524848518918403011753745
7199584002706285601626330852568557111612744053811189951200821882841079611969126136442815811272648576075293843625

All 15 rows

Technical details
Node id
n-2ebf568485516dc1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-778f5ea1bb6e399b

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,          45,           42;
  1:         60,         321,          570,           336;
  2:        360,        2466,         6309,          7146,          3024;
  3:       2520,       20862,        68823,        113112,         92628,         30240;
  4:      20160,      194616,       780066,       1661949,       1985256,       1260828,         332640;
  5:     181440,     1993464,      9355986,      24318333,      37810692,      35170524,       18123696,        3991680;
  6:    1814400,    22293360,    119474892,     364811148,     694245249,     843244236,      638453772,      275524848,      51891840;
  7:   19958400,   270628560,   1626330852,    5685571116,   12744053811,   18995120082,    18828410796,    11969126136,    4428158112,     726485760;
  8:  239500800,  3546918720,  23575398624,   92621816172,  238212212472,  419102248149,   510867730782,   426055675572,  232674789384,   75140200800,   10897286400;
  9: 3113510400, 49941956160, 363230881632, 1581289988220, 4578707820888, 9259724625489, 13346916470550, 13712607474948, 9841663071144, 4699619240544, 1343907936000, 174356582400;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+4+(2*n+11)*x+(n+7)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 45*x + 42*x^2.
%t Mathematica
P[0, x_] := 12 + 45*x + 42*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (2*n + 11)*x + (n + 7)*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 + 45*x + 42*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (2*n + 11)*x + (n + 7)*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.