Real-rooted atlas

A143491 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 10\,x + 6\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 12 + 38\,x + 30\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
012383080
160274416210960
2360212446884588168013440
325201810851932743085305215120215040
420160170064596536111378411674966514801512003870720
5181440175233672395281658595222759088187057768527080166320077414400
618144001970064093423312252733856426622304460166816309740112118956960199584001703116800
7199584002402942401283764752399457547279783854721060792492893893098405335148016176598288025945920040874803200

All 15 rows

Technical details
Node id
n-93bd186d4ea2fd62 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f373be5c339f8575

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,          38,           30;
  1:         60,         274,          416,           210;
  2:        360,        2124,         4688,          4588,          1680;
  3:       2520,       18108,        51932,         74308,         53052,         15120;
  4:      20160,      170064,       596536,       1113784,       1167496,        651480,        151200;
  5:     181440,     1752336,      7239528,      16585952,      22759088,      18705776,       8527080,       1663200;
  6:    1814400,    19700640,     93423312,     252733856,     426622304,     460166816,     309740112,     118956960,      19958400;
  7:   19958400,   240294240,   1283764752,    3994575472,    7978385472,   10607924928,    9389309840,    5335148016,    1765982880,     259459200;
  8:  239500800,  3162948480,  18769296384,   65907612192,  151664682272,  238992495744,  261182667072,  195472113952,   95883866784,   27837270720,   3632428800;
  9: 3113510400, 44710842240, 291445080192, 1138338404256, 2960255052416, 5381872678752, 6980262108096, 6458877487456, 4178571977472, 1800142521120, 464780635200, 54486432000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+4+(2*n+10)*x+(n+6)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 38*x + 30*x^2.
%t Mathematica
P[0, x_] := 12 + 38*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (2*n + 10)*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] = 12 + 38*x + 30*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (2*n + 10)*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.