Real-rooted atlas

A088996 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1011
20134
302111528
40647122105280
502424291115189453640
601201474721717611214231039558240
7072010404624641994873574813408691351351106560
805040836285933082333369549467277482986058980202702524344320
90403207547046168140287531888362464915539580818019290611921886034459425608608000
1003628807558416698526363759733521298932413298743060345742565064496634954257529316565472907517041024000
110362880083204640857253096522663887620884764522571518867031084786077231410257361661201822656846062844721513749310575528271744000
12039916800998713440113434907767720984884434994510389011090203377392507758079122404629107545545655968543423431105030097154569670227031623414322517961239296000
13047900160012982481280161089725312121010545552861295874677802205687165011857818605392939111249444883510155944439138479155313181719714104325986179665424372272754507905853580625664565853952000
1406227020800181705299840244469342361620080793505288112357484380396452238193081594134717740466139330073451290949835031012720654997622957121909521555496937267679233368485581431805124858123298527521345804667687526582634158080000
Technical details
Node id
n-231a68fa99c13c5a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e3fd51a9c1552ae1

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: 0,     1;
  2: 0,     1,      3;
  3: 0,     2,     11,      15;
  4: 0,     6,     47,     122,      105;
  5: 0,    24,    242,     911,     1518,      945;
  6: 0,   120,   1474,    7217,    17611,    21423,     10395;
  7: 0,   720,  10404,   62464,   199487,   357481,    340869,    135135;
  8: 0,  5040,  83628,  593308,  2333369,  5494672,   7748298,   6058980,   2027025;
  9: 0, 40320, 754704, 6168140, 28753188, 83624649, 155395808, 180192906, 119218860, 34459425;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n-1+(3*n-2)*x+(2*n-1)*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 - 1 + (3*n - 2)*x + (2*n - 1)*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 - 1 + (3*n - 2)*x + (2*n - 1)*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

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.