Real-rooted atlas

A038305 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 13\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 9 + 78\,x + 169\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0978169256
127351152121974096
28114049126263642856165536
32435265456301977304284153712931048576
472918954205335118638038557356683274482680916777216
52187663398624076228495269901457017437710136298962748517268435456
66561227448344962829896776161940870561395016121635586815059644088157307214294967296
7196837676371330570813453549287448069837894163581094720281220330519508220247294671060449937368719476736

All 15 rows

Technical details
Node id
n-a8d904c8d696a511 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a39bea8eb66cb9fc

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:      9,      78,       169;
  1:     27,     351,      1521,       2197;
  2:     81,    1404,      9126,      26364,       28561;
  3:    243,    5265,     45630,     197730,      428415,       371293;
  4:    729,   18954,    205335,    1186380,     3855735,      6683274,      4826809;
  5:   2187,   66339,    862407,    6228495,    26990145,     70174377,    101362989,      62748517;
  6:   6561,  227448,   3449628,   29896776,   161940870,    561395016,   1216355868,    1505964408,     815730721;
  7:  19683,  767637,  13305708,  134535492,   874480698,   3789416358,  10947202812,   20330519508,   22024729467,   10604499373;
  8:  59049, 2558790,  49896405,  576580680,  4372403490,  22736498148,  82104021090,  203305195080,  330370942005,  318134981190,  137858491849;
  9: 177147, 8444007, 182953485, 2378395305, 20612759310, 125050739814, 541886539194, 1677267859410, 3634080362055, 5249227189635, 4549330231017, 1792160394037;
%F Formula
The row polynomials satisfy P(n,x) = (3+13*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 78*x + 169*x^2.
%t Mathematica
P[0, x_] := 9 + 78*x + 169*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 13*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] = 9 + 78*x + 169*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 13*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.