Real-rooted atlas

A225470 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
2185440112
31626849544402240
41944104762102418636616058240
5291601901884934526369484092121047201863680
6524880400658412685896213341042010477610069200209440070819840
71102248096210504358555248739791792912884688673863048275574000481712003116072960
82645395202595636576111067990562707743945641143819104399077150402413421524883210328001252451200155803648000
97142567040777538336323751570352161053188037936189612886003222706790600961808947547856924560127792275126133600363210848008725004288000
102142770112002561177154240137428337327044360066626499290585883014912128796495323904126930156358752856231253651523783970809734498936688192001162274713600540950265856000
1170711413696009201854148192054315471357763219198211673893764515357458766816742079025121075286965724961754567268118609606336424551975499267216508808544406403846334742208004067961497600036784618078208000
1225456108930560035813708653939202305027426510771289753441141967552235506072876401664438732032476726080595066639408324992592122024800495424429027218344776960220761461449584576765802775406931201608053815952640015458253690880002722061737787392000
1399278824829184001501104684118924801045796901820351488444544544940615072012864627929000334528267662982545247853444119561226847044396847490491182960646208410090645322666138242619981294866965382412037850743519999296376693236738994752071958925393501440063378840132608000217764939022991360000
1441697106428257280067414665025478937605052833048657803161623272377255515379571273591397279188468185616902281555660557036802907932838474849201792380720756949704667532838119623224054662137602904790983263856468864165838150096930473868868787659214525776486419596777283042829568034323838458710169600278866896583475200018727784755977256960000
Technical details
Node id
n-9128a95b5b9242bf · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-67f6b8af5e681e1c

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,           5;
  2:         18,          54,           40;
  3:        162,         684,          954,           440;
  4:       1944,       10476,        21024,         18636,          6160;
  5:      29160,      190188,       493452,        636948,        409212,        104720;
  6:     524880,     4006584,     12685896,      21334104,      20104776,      10069200,       2094400;
  7:   11022480,    96210504,    358555248,     739791792,     912884688,     673863048,     275574000,     48171200;
  8:  264539520,  2595636576,  11106799056,   27077439456,   41143819104,   39907715040,   24134215248,   8321032800,   1252451200;
  9: 7142567040, 77753833632, 375157035216, 1053188037936, 1896128860032, 2270679060096, 1808947547856, 924560127792, 275126133600, 36321084800;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3*n+(6*n+2)*x+(3*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[((3*n + (6*n + 2)*x + (3*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] = ((3*n + (6*n + 2)*x + (3*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

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.