Real-rooted atlas

A272866 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 15\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1915529
28127031515025841
37293645729074254050112512524389
4656143740123930194400184275108000382507500625707281
55904949207518042753827250519412547081252885625118125030937546875312520511149
653144153144102391484563969750113177250139421250122563125774562503493125010968750227812528125015625594823321
7478296955801305297606960961022475209771572532722987503760273125323265937520890406251009968750359690625915468751575000016406257812517249876309

All 15 rows

Technical details
Node id
n-2406581cbd276ab3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-970ef418506da5d4

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         9,         15,           5;
  2:        81,        270,         315,          150,           25;
  3:       729,       3645,        7290,         7425,         4050,          1125,           125;
  4:      6561,      43740,      123930,       194400,       184275,        108000,         38250,          7500,           625;
  5:     59049,     492075,     1804275,      3827250,      5194125,       4708125,       2885625,       1181250,        309375,         46875,          3125;
  6:    531441,    5314410,    23914845,     63969750,    113177250,     139421250,     122563125,      77456250,      34931250,      10968750,       2278125,       281250,        15625;
  7:   4782969,   55801305,   297606960,    961022475,   2097715725,    3272298750,    3760273125,    3232659375,    2089040625,    1009968750,     359690625,     91546875,     15750000,      1640625,       78125;
  8:  43046721,  573956280,  3539397060,  13392313200,  34782813450,   65721537000,   93415518000,  101859525000,   86092621875,   56588625000,   28831950000,  11269125000,   3313406250,    708750000,   104062500,    9375000,    390625;
  9: 387420489, 5811307335, 40679151345, 176491556100, 531627004350, 1180197600750, 2000476784250, 2646576180000, 2769804061875, 2309984578125, 1538780034375, 816844500000, 343017281250, 112425468750, 28134843750, 5189062500, 664453125, 52734375, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (9+15*x+5*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(9 + 15*x + 5*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (9 + 15*x + 5*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.