Real-rooted atlas

A136572 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
22162442
3672216192486
4243841728307219207128
5120240014400384004800023040126360
6720172801296004608008640008294403225602624400
7504014112012700805644800141120002032128015805440516096062460720
84032012902401354752072253440225792000433520640505774080330301440928972801675416960
9362880130636801567641609754214403657830400877879296013655900160133772083207524679680185794560049997969280
10362880014515200019595520001393459200060963840000175575859200341397504000445906944000376233984000185794560000408748032001642790419200
1139916800175633920026345088000210760704000105380352000035407798272008261819596800134886850560001517477068800011240570880000494585118720098099527680058926178080000
124790016002299207680037936926720033721712640001896846336000072838899302400198283670323200388474129612800546291744768000539547402240000356101285478400141263319859200255058771968002291049803750400
136227020800323805081600582849146880056989694361600356185589760000153872174776320047871343263744001094202131742720018464660973158400227958777446400002006037241528320011936750528102400431049324625920071416456151040095969530668211200
14871782912004881984307200951986939904001015452735897600698123755929600033509940284620800117284790996172800306376596887961600603178925123174400893598407589888000982958248348876800779867701169356800422428338133401600139976254056038400214249368453120004308700997586585600
Technical details
Node id
n-0058416e6064cb2b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7cabb77b45806852

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:      1,        4;
  2:      2,       16,        24;
  3:      6,       72,       216,       192;
  4:     24,      384,      1728,      3072,       1920;
  5:    120,     2400,     14400,     38400,      48000,      23040;
  6:    720,    17280,    129600,    460800,     864000,     829440,      322560;
  7:   5040,   141120,   1270080,   5644800,   14112000,   20321280,    15805440,     5160960;
  8:  40320,  1290240,  13547520,  72253440,  225792000,  433520640,   505774080,   330301440,   92897280;
  9: 362880, 13063680, 156764160, 975421440, 3657830400, 8778792960, 13655900160, 13377208320, 7524679680, 1857945600;
%F Formula
The row polynomials satisfy (1+2*n*x)*P(n,x) = (n+(2*n^2+4*n)*x+(4*n^2+4*n)*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*n^2 + 4*n)*x + (4*n^2 + 4*n)*x^2)*P[n - 1, x])/(1 + 2*n*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*n^2 + 4*n)*x + (4*n^2 + 4*n)*x^2)*v[i - 1])/(1 + 2*n*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

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.