Real-rooted atlas

A038284 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 54\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 3456\,x + 2916\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01024345629167396
132768165888279936157464636056
2104857670778881791590420155392850305654700816
333554432283115520955514880161243136013604889604591650244704270176
4107374182410871635968458647142401031956070401306069401608815968460824794911296404567235136
53435973836840587440947220547391979525778953994240975198486528098738846760965554060130304133892520998434792782221696
610995116277761484340697497687668872445952295882444505088624127031377920842571492360192710919696678912342764853755904723019613391362992179271065856
73518437208883253436265109913636069478949191681420235733624422435949717007368192606651474499338246824829088117555249358138940850176208229648656711683904305912313344257327417311663616

All 15 rows

Technical details
Node id
n-d4c4d98e7be818e1 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-50db7ebeb5f2c85c

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:              1024,               3456,                2916;
  1:             32768,             165888,              279936,               157464;
  2:           1048576,            7077888,            17915904,             20155392,              8503056;
  3:          33554432,          283115520,           955514880,           1612431360,           1360488960,             459165024;
  4:        1073741824,        10871635968,         45864714240,         103195607040,         130606940160,           88159684608,           24794911296;
  5:       34359738368,       405874409472,       2054739197952,        5778953994240,        9751984865280,         9873884676096,         5554060130304,         1338925209984;
  6:     1099511627776,     14843406974976,      87668872445952,      295882444505088,      624127031377920,       842571492360192,       710919696678912,       342764853755904,        72301961339136;
  7:    35184372088832,    534362651099136,    3606947894919168,    14202357336244224,    35949717007368192,     60665147449933824,     68248290881175552,     49358138940850176,     20822964865671168,      3904305912313344;
  8:  1125899906842624,  18999560927969280,  144277915796766720,   649250621085450240,  1917318240392970240,   3882569436795764736,   5459863270494044160,   5264868153690685440,   3331674378507386880,   1249377891940270080,   210832519264920576;
  9: 36028797018963968, 668784544664518656, 5642869595606876160, 28567027327759810560, 96413717231189360640, 227777406958684864512, 384374374242780708864, 463308397524780318720, 390916460411533393920, 219890508981487534080, 74213046781252042752, 11384956040305711104;
%F Formula
The row polynomials satisfy P(n,x) = (32+54*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 3456*x + 2916*x^2.
%t Mathematica
P[0, x_] := 1024 + 3456*x + 2916*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 54*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] = 1024 + 3456*x + 2916*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 54*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.