Real-rooted atlas

A196347 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(2\,n + 8\,n x + 8\,n x^{2} + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
2409632168
3336144017283843888
434562150446080368646144114048
542240345600107520015360009216001228804043520
65990406082560248832005160960055296000265420802949120167961600
79676800117411840596090880162570240025288704002167603200867041280825753607994972160
81754726402477260800150287155205086642176010404495360012947816448092484403200317089382402642411520428906741760
93530096640568531353604013162496001623101276160412018016256067421129932806991820881920428070666240012842119987209512681472025598960271360
1078033715200141203865600011370627072000535088332800001623101276160003296144130048004494741995520003995326218240002140353331200005707608883200038050725888001682217389260800
111880240947200377683181568003417133547520001834461167616000647456882688000015711620353228800265888959823872003107793036902400024171723620352000115103445811200002762482699468800167423193907200120680812707840000
124904976384000010830187855872001087727562915840065608964112384000264162408136704000745870328856576000150831555390996480021878862979792896002237610986569728000154699031170252800066299584787251200014465363953582080080363133075456009384139996161638400
131377317368627200331576403558400003660603495284736002451012775103692800110879149349928960003571475758008238080084034723717840896000145660187777590886400184876392179249971200168069447435681792000104576545071090892800407441084692561920008148821693851238400417888291992371200786182395233986150400
144142154456760320010798168170037248001299779501948928000095663771343441100800480398503920323788800173858506180688609280046667283237974310912009411889056398180352000142746984022039068672001610478794094799749120013176644678957452492800745345557597593272320026619484199914045440004914366313830285312002340174435157278720070593757144458618470400
Technical details
Node id
n-6e77991891ce5749 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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:          6,           4;
  2:         40,          96,           32;
  3:        336,        1440,         1728,           384;
  4:       3456,       21504,        46080,         36864,          6144;
  5:      42240,      345600,      1075200,       1536000,        921600,        122880;
  6:     599040,     6082560,     24883200,      51609600,      55296000,      26542080,       2949120;
  7:    9676800,   117411840,    596090880,    1625702400,    2528870400,    2167603200,     867041280,      82575360;
  8:  175472640,  2477260800,  15028715520,   50866421760,  104044953600,  129478164480,   92484403200,   31708938240,    2642411520;
  9: 3530096640, 56853135360, 401316249600, 1623101276160, 4120180162560, 6742112993280, 6991820881920, 4280706662400, 1284211998720, 95126814720;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (4*n^2+2*n+(8*n^2+8*n)*x+8*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[((4*n^2 + 2*n + (8*n^2 + 8*n)*x + 8*n*x^2)*P[n - 1, x])/(2*n - 1 + 2*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] = ((4*n^2 + 2*n + (8*n^2 + 8*n)*x + 8*n*x^2)*v[i - 1])/(2*n - 1 + 2*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.