Real-rooted atlas

A145324 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2} + n + 4\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13710
2125870140
36044611049102520
4360363613760231241456055440
52520322921654044233085412762766401441440
620160313776203365670253521364298414121192608608043243200
71814403327984261473041140695682984206564681653284078045201521520001470268800
818144003836016035465659218728201926178154464130374316481718667438412940046560426025600055870214400
9199584004782081605090387472315953764641260171250563349345365125932137993126751274180644480042593601320679360002346549004800
1023950080064170835207734372710455221769161625864483004488303796690048185063398331522827079819337628330383326496168169600502404490309824000107941254220800
11311351040092283615360124290054259210040547893856540558824956162036479440872005478228953627521052254950340512141431451639936012668446637334726806015495708801661414634880005397062711040000
1243589145600141651103104021091952210688190283692217664115840427069286450133065170454401581543829856652836644485119277248618906012432115207430840594824300860202208243331200295500424716672006645658539520000291441386396160000
136538371840002312199872640037728925749504037612093283245442555826282575251212501098139547475245280375471145184012297311236275194242504071878777094464377592194268174048040982945794244173443031945591538249600137033670437448960028576331719936000016903600410977280000
1410461394944000400028490086400710024006133504077534655097964544581947834314969216317585579231221158412995365219575068032405046706947670954249663278174729940492817560205750665419302423926512263415073958423703268011813519142416139488448075131520067607701476416281600131451125911705600001048023225480591360000
Technical details
Node id
n-e0e020c9a0b1ac7f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-588beb345c2d0d0e

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,         7;
  2:       12,        58,         70;
  3:       60,       446,       1104,         910;
  4:      360,      3636,      13760,       23124,        14560;
  5:     2520,     32292,     165404,      423308,       541276,       276640;
  6:    20160,    313776,    2033656,     7025352,     13642984,     14121192,      6086080;
  7:   181440,   3327984,   26147304,   114069568,    298420656,    468165328,    407804520,    152152000;
  8:  1814400,  38360160,  354656592,  1872820192,   6178154464,  13037431648,  17186674384,  12940046560,   4260256000;
  9: 19958400, 478208160, 5090387472, 31595376464, 126017125056, 334934536512, 593213799312, 675127418064, 448004259360, 132067936000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+2+(4*n+6)*x+(3*n+4)*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 + (4*n + 6)*x + (3*n + 4)*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 + (4*n + 6)*x + (3*n + 4)*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.