Real-rooted atlas

A139548 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(18 + 12\,x + 2\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11812232
23244322164841024
3583211664972043201080144832768
410497627993632659221772890720241924032384161048576
5188956862985609447840839808048988801959552544320103680129609603233554432
63401222413604889624942297627713664020785248011085465643110144123171842566080380160380162304641073741824
761222003228570268166190224768825363302475658302725043886848252194342496074035228021593662270208103783681257984104832537612834359738368

All 15 rows

Technical details
Node id
n-a046a8a92333afa9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-12929af9fbcc0757

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:           18,            12,             2;
  2:          324,           432,           216,            48,             4;
  3:         5832,         11664,          9720,          4320,          1080,           144,             8;
  4:       104976,        279936,        326592,        217728,         90720,         24192,          4032,           384,            16;
  5:      1889568,       6298560,       9447840,       8398080,       4898880,       1959552,        544320,        103680,         12960,          960,           32;
  6:     34012224,     136048896,     249422976,     277136640,     207852480,     110854656,      43110144,      12317184,       2566080,       380160,        38016,        2304,         64;
  7:    612220032,    2857026816,    6190224768,    8253633024,    7565830272,    5043886848,    2521943424,     960740352,     280215936,     62270208,     10378368,     1257984,     104832,       5376,       128;
  8:  11019960576,   58773123072,  146932807680,  228562145280,  247608990720,  198087192576,  121053284352,   57644421120,   21616657920,   6404935680,   1494484992,   271724544,   37739520,    3870720,    276480,    12288,    256;
  9: 198359290368, 1190155742208, 3372107936256, 5994858553344, 7493573191680, 6994001645568, 5051223410688, 2886413377536, 1322939464704, 489977579520, 146993273856, 35634733056, 6928975872, 1065996288, 126904320, 11280384, 705024, 27648, 512;
%F Formula
The row polynomials satisfy P(n,x) = (18+12*x+2*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[(18 + 12*x + 2*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] = (18 + 12*x + 2*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.