Real-rooted atlas

A088996 → basis A013610 → basis A007318

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
1437
2365118105
350410387111622415
49576257702596511610194474865
52298247621201009710668115220806291602919735
66664896262383124299975037550115184294444820148524880137227545
72266064641032065424201299605221796986601415153511550903356119069028110224807547514975
888376520964568910667210327641620413332015298810750375476955448915148178653726363287533392264539520475493443425
9388856692224224893730016057777681931208654549968980832980934051253423536452752283197104980627016530420100403140464714256704033760034483175
1019053977918976121863628474560350578760467680597405994273620667797156754488511669608833835272147267502270992197231399802373024979533633620799988802142770112002667042724170825
111028914807624704720941720895244822952752804915200438290227862089205577544427577181249667465049925194315810495366391351433872487713383045556843525674849646505631856081225195985678407071141369600232032717002861775
126070597364985753646239654840268377616137514350122849283412211445363273480486859603257405802849382964318733716783651310664459015949198290255106990483078497947237829943622091901992038029641956076590184448482790288924864025456108930560022043108115271868625
133885182313590882304319609120701162055681213135572284909026562813178384687286149124446663924539073795125059262169102840504844262774433684385161782693068791823755311311275718857239374522744475301669765801694811301030678666636216194527277192910643220458012239625536099278824829184002270440135873002468375
142680775796377708789762368480590008835240960971299375571075291712024506100257938892340480424973302950062110126325358489744987370942290050662044702654187736610364858272850583293275152011334904061145651440884459773525437761649202659397818129634610720608867109767107064880958174848663640899849277389031964094720416971064282572800252018855081903273989625
Technical details
Node id
n-fe9a040afa3cc625 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d5423eb916062c66

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:            4,             3;
  2:           36,            51,            18;
  3:          504,          1038,           711,           162;
  4:         9576,         25770,         25965,         11610,          1944;
  5:       229824,        762120,       1009710,        668115,        220806,         29160;
  6:      6664896,      26238312,      42999750,      37550115,      18429444,       4820148,        524880;
  7:    226606464,    1032065424,    2012996052,    2179698660,    1415153511,     550903356,     119069028,     11022480;
  8:   8837652096,   45689106672,  103276416204,  133320152988,  107503754769,   55448915148,   17865372636,   3287533392,    264539520;
  9: 388856692224, 2248937300160, 5777768193120, 8654549968980, 8329809340512, 5342353645275, 2283197104980, 627016530420, 100403140464, 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.