Real-rooted atlas

A038226 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 704\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
010247041211849
1327683379211616133179507
210485761441792743424170368146413418801
3335544325767168039649280136294402342560161051147008443
41073741824221459251219031654408722841602248857603092179217715616321363049
53435973836882678120448852618117124884791296016791470080346324070439682966419487171271818611107
61099511627776302365697638436378372997122501013143552107465408512029552987340850794196992498871577621435888111688200277601
73518437208883210885165114982414967102033100812004863089049661900075302912212781508853764876242911232718375071744617353577282357947691502592611936843

All 15 rows

Technical details
Node id
n-b3ac70975a2c0d1d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e63d418fb4410416

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,                704,                121;
  1:             32768,              33792,              11616,               1331;
  2:           1048576,            1441792,             743424,             170368,              14641;
  3:          33554432,           57671680,           39649280,           13629440,            2342560,            161051;
  4:        1073741824,         2214592512,         1903165440,          872284160,          224885760,          30921792,           1771561;
  5:       34359738368,        82678120448,        85261811712,        48847912960,        16791470080,        3463240704,         396829664,         19487171;
  6:     1099511627776,      3023656976384,      3637837299712,      2501013143552,      1074654085120,      295529873408,       50794196992,       4988715776,        214358881;
  7:    35184372088832,    108851651149824,    149671020331008,    120048630890496,     61900075302912,    21278150885376,     4876242911232,     718375071744,      61735357728,      2357947691;
  8:  1125899906842624,   3870280929771520,   5986840813240320,   5487937412136960,   3301337349488640,  1361801656664064,   390099432898560,   76626674319360,    9877657236480,    754543261120,   25937424601;
  9: 36028797018963968, 136233888727957504, 234151996251176960, 241469246134026240, 166010106717143040, 79892363857625088, 27463000076058624, 6743147340103680, 1158978449080320, 132799613957120, 9129973459552, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (32+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 704*x + 121*x^2.
%t Mathematica
P[0, x_] := 1024 + 704*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 11*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 + 704*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 11*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.