Real-rooted atlas

A038259 → basis A013610 → D

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(11\,n + 15\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 15\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11515
2330450780
35445148501012530420
4798603267004455002025001054560
5109807559895001225125011137500379687534273200
61449459098826750269527500367537500250593750683437501069323840
718601390515219319505188404375943346250096478593755262468750119601562532436156480
82338460520223216686009131591700020753617500028300387500023154862500010524937500020503125000963817221120
92893844893531569217020015067126305004109216265000700434590625076411046250005209844062500202980937500034599023437528191653717760
1035369215365043407673402502367691276500075335631525000154095609937500210130377187500191027615625000111639515625000380589257812505766503906250814425551846400
114279675059165583592053522503581133055706251302230202075000310759480040625050851551279375005778585372656250450279379687500023025650097656256977469726562509514731445312523292570782807040
12513561007099807703415106497005252328481702500214867983342375005860035909337500011187341281462500015255465383812500014859219529687500010131286042968750046051300195312500125594455078125001556956054687500660662007657799680
1361199353346059510014439638446100751082972883457503414013513106625001047481418794078125228541400464162500036358859164753125004249736785490625000362193476036132812521951119759765625008980003538085937502226447158203125002530053588867187518608646549028024320
147249769550225510128518642026724950105151616203684050052575808101842025001792357094380978125043994219589351281250799894901624568750001090765774942593750001115555906191289062508451181107509765625046097351495507812500171436431181640625003896282526855468750408700964355468750521042103372784680960
Technical details
Node id
n-01ef370b792e48e5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c62c9860d860e562

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:           0;
  1:          15;
  2:         330,          450;
  3:        5445,        14850,         10125;
  4:       79860,       326700,        445500,        202500;
  5:     1098075,      5989500,      12251250,      11137500,       3796875;
  6:    14494590,     98826750,     269527500,     367537500,     250593750,      68343750;
  7:   186013905,   1521931950,    5188404375,    9433462500,    9647859375,    5262468750,    1196015625;
  8:  2338460520,  22321668600,   91315917000,  207536175000,  283003875000,  231548625000,  105249375000,   20503125000;
  9: 28938448935, 315692170200, 1506712630500, 4109216265000, 7004345906250, 7641104625000, 5209844062500, 2029809375000, 345990234375;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (11*n+15*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 15.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 15;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 15*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 15; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 15*n*x)*v[i - 1])/(n - 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

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.