Real-rooted atlas

A143491 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 2\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 30 + 22\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
03022456
1210214728504
2168021321004208165040
315120225481330038805603255440
41512002557201780966540013360144064665280
516632003115320247049610755922777604256035841288648640
619958400407102403587659217848096548430410662401281288704256121080960
7259459200569149920547816176303778432106992144248297283798144369408207365121816214400
83632428800848701728088077263045348530400210544688056160048010283347212768000102912048640102429059430400
95448643200013457011680014908992912097843408608422787640001263490096026657030403971869444097280027878401126402048494010316800
1087178291200022620947328002654579099520186367439596887214704121628671594336067921050560116863971841449938688126551040737792025804840968892185702400
11148203095040004019917628160049652034157440369916229304961855384849260866184651195521728089746240334510853248480217520645051245056378526720191426565857288192168951528345600
122667655710720007532257920768009741349673971207651532810638084079525187279361562400691371524434254567142494773748509441533413243648186965915136169159710721101621248488284161318912163843379030566912000
13506854585036800014844821191603200200150159646988801648618227500659292814044179584003784466351061760115498850603136026875521351078448089601331200661917887488069533528064054762745856313098240012271616029491203276870959641905152000
1410137091700736000030703351553280000042998996167718400036975367742952960021860045290918118494252135857152000306687028227507207685081282278400149930245364556822856278016000027145063362560248592547840017214513971287162880003044147206553600655361561112121913344000
Technical details
Node id
n-78add49ce830f310 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f01da5544602b4b6

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:          30,           22,            4;
  1:         210,          214,           72,           8;
  2:        1680,         2132,         1004,         208,          16;
  3:       15120,        22548,        13300,        3880,         560,          32;
  4:      151200,       255720,       178096,       65400,       13360,        1440,         64;
  5:     1663200,      3115320,      2470496,     1075592,      277760,       42560,       3584,       128;
  6:    19958400,     40710240,     35876592,    17848096,     5484304,     1066240,     128128,      8704,      256;
  7:   259459200,    569149920,    547816176,   303778432,   106992144,    24829728,    3798144,    369408,    20736,     512;
  8:  3632428800,   8487017280,   8807726304,  5348530400,  2105446880,   561600480,  102833472,  12768000,  1029120,   48640,   1024;
  9: 54486432000, 134570116800, 149089929120, 97843408608, 42278764000, 12634900960, 2665703040, 397186944, 40972800, 2787840, 112640, 2048;
%F Formula
The row polynomials satisfy P(n,x) = (n+6+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 30 + 22*x + 4*x^2.
%t Mathematica
P[0, x_] := 30 + 22*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 6 + 2*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] = 30 + 22*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 6 + 2*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.