Real-rooted atlas

A034870 → D → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27Pn(1)
00
1426
23248244108
31924804802406061458
410243584537644802240672112817496
5512023040460805376040320201606720144018010196830
62457613516833792050688050688035481617740863360158402640264122125764
71146887454722236416410009651251204612608307507215375365765761601603203243683641422320522
85242883932160137625602981888044728320492011524100096026357760131788805125120153753634944058240672048016229582512
92359296200540168021606420054016035094528045622886445622886435846553622404096011202048044808192142571523564288685440979209792612182324522934
10104857609961472044826624012700876802540175360381026304044453068804127784960309583872018919014409459507203869798401289932803472896074419201240320155040136807602023245229340
11461373444844421122422210560767033344017258250240293390254083911870054441912893440366737817602648662016015891972096794598604833108275201146055680327444480764037121432569621067202340801848092422230127770466
12201326592231525580812733906944445686743041114216857602117012029443175518044163855986196483855986196483213321830402249325281281329146757126645733785628116566016100416307203012489216753122304155054592258424323400320340032242881104242259436291848
138724152321090519040065431142400250819379200689753292800144848191488024141365248003276328140800368586915840034810986496002784878919680189878108160011076222976005538111488002373476352008702746624027196083200719896320015997696002946944004420416052624004784003120013002622029503845518
14375809638450734301184329772957696137405399040041221619712009480972533760173817829785602607267446784032590843084800344014454784003096130093056023924641628160159497610854409201785241600460089262080019937201356807476450508802418851635206719032320015913497600318269952053044992072334080786240065520039312151228213516729579636
Technical details
Node id
n-91fbe90ea8777148 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-177c65847e3c5511

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • 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:       0;
  1:       4,        2;
  2:      32,       48,       24,         4;
  3:     192,      480,      480,       240,        60,         6;
  4:    1024,     3584,     5376,      4480,      2240,       672,       112,         8;
  5:    5120,    23040,    46080,     53760,     40320,     20160,      6720,      1440,       180,        10;
  6:   24576,   135168,   337920,    506880,    506880,    354816,    177408,     63360,     15840,      2640,      264,       12;
  7:  114688,   745472,  2236416,   4100096,   5125120,   4612608,   3075072,   1537536,    576576,    160160,    32032,     4368,     364,     14;
  8:  524288,  3932160, 13762560,  29818880,  44728320,  49201152,  41000960,  26357760,  13178880,   5125120,  1537536,   349440,   58240,   6720,   480,   16;
  9: 2359296, 20054016, 80216064, 200540160, 350945280, 456228864, 456228864, 358465536, 224040960, 112020480, 44808192, 14257152, 3564288, 685440, 97920, 9792, 612, 18;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+4*n*x+n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 4*n*x + n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 4*n*x + n*x^2)*v[i - 1])/(n - 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.