Real-rooted atlas

A132393 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
22101224
36408660192
4241965847563601920
51201148429278687092252023040
6720784834936815441054967185620160322560
7504061416315184885232147236814524567878241814405160960
8403205417283135632102336962063126426343328208271529329760181440092897280
9362880531907234179696126595216298252032464033856477220976313066512118956960199584001857945600
103628800575452804056258241676108512450166291282193629441034061603288573168324946367744162706752023950080040874803200
11399168006801724805209972704237103293447130770268814893461024022059849462423185849356816955516400082200523392237863865603113510400980995276800
1247900160087209049607204208716835746356998411876370430722785523160512473226647884858706808475525280680877952336017857670414362439662083703715366404358914560025505877196800
13622702080012055678848010673607075845727657717312208012351096965402635673273610330231163270414730284820089615670906412665612289253466643269073850211264263584894694406122231942400653837184000714164561510400
14871782912001787427371520168719585218569726497936371238285981501273610891887560134402310654070581632371507686093580845507724689875204227840511356544293331445762060814742004559523845074470787046401071094316544001046139494400021424936845312000
Technical details
Node id
n-bc8d19a26ba87db3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-56ff70451877aec3

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,       3;
  2:      2,      10,       12;
  3:      6,      40,       86,        60;
  4:     24,     196,      584,       756,       360;
  5:    120,    1148,     4292,      7868,      7092,      2520;
  6:    720,    7848,    34936,     81544,    105496,     71856,     20160;
  7:   5040,   61416,   315184,    885232,   1472368,   1452456,    787824,    181440;
  8:  40320,  541728,  3135632,  10233696,  20631264,  26343328,  20827152,   9329760,   1814400;
  9: 362880, 5319072, 34179696, 126595216, 298252032, 464033856, 477220976, 313066512, 118956960, 19958400;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+(2*n+2)*x+(n+2)*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)*x + (n + 2)*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*n + 2)*x + (n + 2)*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.