Real-rooted atlas

A120910 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(7 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
1213657
21475044321083
3102952929072518420577
472034939212700814515262208390963
5504214321801481760254016021772807464967428297
63529473630312155584803556224045722880313528328957952141137643
72470629296475481524731044356374407468070407681443844389396481074954242681615217

All 15 rows

Technical details
Node id
n-9f4aefa5f442e451 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-48a0cbb267bcb941

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:         3;
  1:        21,         36;
  2:       147,        504,         432;
  3:      1029,       5292,        9072,        5184;
  4:      7203,      49392,      127008,      145152,        62208;
  5:     50421,     432180,     1481760,     2540160,      2177280,       746496;
  6:    352947,    3630312,    15558480,    35562240,     45722880,     31352832,      8957952;
  7:   2470629,   29647548,   152473104,   435637440,    746807040,    768144384,    438939648,    107495424;
  8:  17294403,  237180384,  1423082304,  4879139328,  10455298560,  14338695168,  12290310144,   6019743744,  1289945088;
  9: 121060821, 1867795524, 12807740736, 51230962944, 131736761856, 225834448896, 258096513024, 189621927936, 81266540544, 15479341056;
%F Formula
The row polynomials satisfy P(n,x) = (7+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(7 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = (7 + 12*x)*v[i - 1]); 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.