Real-rooted atlas

A059364 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(1 + 5\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
2666918153
3105616989091623825
4221764833039465143101944126225
557657615892201751040964035265194291605175225
617873856596441888288820061403805255736445677452524880253586025
764345881625225417444236503148395118918022101310897414347961381221721102248014454403425
82638181145611886722308823423763092426367483193218544391496983441972772234574441563766853808264539520939536222625
91213563326976618020117136013984346045880184534583038201564964055073888453164516753331975696020806626267380113873870736714256704068586144251625
1061891729675776351597159548640898607683480680136065675487122013517354172022389206003555575654352902540472701410972105169803000635542893637804870411202142770112005555477684381625
1134659368618434562173186801402444861924736540023200105850831827650760120598856274075588961608886788974945475606596004676522266022172510790633656385108075612019170034576081367555849841607071141369600494437513909964625
12211422148572450816145041767588185574445597561774462953288686191256927531560111671601785140382281020737303527946830268019120160031624493329445727084841730118810719312631455630143323584982130451611102808507648535454068297536025456108930560047960438849266568625
1313953861805781753856103973030402528060928357510197070847865664751119113877622600752107579403080210025388811092058672904923988248470137414321079854124850179866117228896912082634581026455881666623077409801447374815162228983504535624236623269460817203222562811853020864099278824829184005035846079172989705625
1499072418821050452377679681473544223259878402975009126893637702112068344885362286815011280107928378969809267257632123936965871313170979800106724622067880347714860700108541895808605553653515746096298019812680813449450201270142339080385821076994542992672080481613862437071048011540136958918803878410181260634555969280416971064282572800569050606946547836735625
Technical details
Node id
n-36d3521195568434 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7df42a29557eb2b9

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,             3;
  2:            66,            69,             18;
  3:          1056,          1698,            909,            162;
  4:         22176,         48330,          39465,          14310,           1944;
  5:        576576,       1589220,        1751040,         964035,         265194,         29160;
  6:      17873856,      59644188,       82888200,       61403805,       25573644,       5677452,        524880;
  7:     643458816,    2522541744,     4236503148,     3951189180,     2210131089,     741434796,     138122172,     11022480;
  8:   26381811456,  118867223088,   234237630924,   263674831932,   185443914969,   83441972772,   23457444156,   3766853808,    264539520;
  9: 1213563326976, 6180201171360, 13984346045880, 18453458303820, 15649640550738, 8845316451675, 3331975696020, 806626267380, 113873870736, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+1+3*n*x)*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[(5*n + 1 + 3*n*x)*P[n - 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] = (5*n + 1 + 3*n*x)*v[i - 1]); 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.