Real-rooted atlas

A145324 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 5\,x + 3\,x^{2} + n + 3\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
212413588
3603135443151232
4360253867077874346520944
52520224467994314230912661745045418880
6201602173689762342337617314757122596156756759634240
71814402299032124813623763453168067628738452424449453011486475250490240
81814400264376801684952286134911881395732369203173735218480048989602608202182430257264216960
91995840032891688024086387881028680285628236371007516594906216299453827049370971878225661504954583103525232454942720
1023950080044060457603646875369617884032639657543291777212693504206131944102743523204132604274614063261591345740187036851054113810758135922995200
113113510400632661148805842449420483236643085548119516360909363088737841226957007096181124751398071437736931539113739242620397126255157208155461002635284526875309165073817600
1243589145600969790389120988761429043261087616632968254712268582900755117472227038163205856366699925911488969031702999190268805359246820112047715413708421400542854613560031209507115268222562512675768026521600
1365383718400015810941059200176438135640960120305506391704855922249110995721871341787230967046379285149589087861969317998905211201311720422723151239995346025127219214046459465174146674762108388515144466143238919252063427784543125557733793166950400
141046139494400027324400965120033131493430905602471846322754252812677030555902164047277365820104145213221845164350251622816908748435510033459420363847296319157080588869506453015318233006192322207360315059616842021116780634545423147654808598885933396756396626132083687526213488278846668800
Technical details
Node id
n-55e57b30e381e5d9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b5725c9059b1c9be

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:        3,         5;
  2:       12,        41,         35;
  3:       60,       313,        544,         315;
  4:      360,      2538,       6707,        7874,        3465;
  5:     2520,     22446,      79943,      142309,      126617,       45045;
  6:    20160,    217368,     976234,     2337617,     3147571,     2259615,      675675;
  7:   181440,   2299032,   12481362,    37634531,    68067628,    73845242,    44494530,    11486475;
  8:  1814400,  26437680,  168495228,   613491188,  1395732369,  2031737352,  1848004898,   960260820,   218243025;
  9: 19958400, 328916880, 2408638788, 10286802856, 28236371007, 51659490621, 62994538270, 49370971878, 22566150495, 4583103525;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+2+(3*n+5)*x+(2*n+3)*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[Together[((n + 2 + (3*n + 5)*x + (2*n + 3)*x^2)*P[n - 1, x])/(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] = ((n + 2 + (3*n + 5)*x + (2*n + 3)*x^2)*v[i - 1])/(1 + x)); 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

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.