Real-rooted atlas

A013620 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 9\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
199
290162252
3675243021875292
4450024300437402624498784
5281252025005467506561002952451728720
61687501518750546750098415008857350318864629042496
7984375106312504784062511481750015500362511160261033480783474360768
856250007087500038272500011481750002066715000223205220013392313203443737687589772288
931640625455625000287043750010333575000232505437503348078300030132704700154968195603486784401119538913536
1017578125028476562502050312500086113125000232505437500418509787500502211745000387420489000174339220050348678440101859494210560
119667968751740234375014095898437567660312500021312998437504603607662500690541149375071027089650004794328551375191773142055034519165569928636210842624
12527343750010441406250093972656250050745234375001826828437500046036076625000828649379250001065406344750009588657102750057531942616500207114993419403389154437772437353038323712
1328564453125616992187500610822265625036649335937500148429810546875427477854375000897703494187500138502824817500015581567791968751246525423357500673123728613050220295038455180330442557682776633187747909632
141538085937503599121093750388705078125002565453515625001154454082031250374043122578125089770349418750001615866289537500021814194908756250218141949087562501570622033430450077103263459313002313097903779390320275094369454100008061430022144
Technical details
Node id
n-dff5aa7e3bad489c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-dca555f310728b4d

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:        9;
  2:       90,       162;
  3:      675,      2430,       2187;
  4:     4500,     24300,      43740,       26244;
  5:    28125,    202500,     546750,      656100,      295245;
  6:   168750,   1518750,    5467500,     9841500,     8857350,     3188646;
  7:   984375,  10631250,   47840625,   114817500,   155003625,   111602610,    33480783;
  8:  5625000,  70875000,  382725000,  1148175000,  2066715000,  2232052200,  1339231320,   344373768;
  9: 31640625, 455625000, 2870437500, 10333575000, 23250543750, 33480783000, 30132704700, 15496819560, 3486784401;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+9*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 9.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 9;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 9*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] = 9; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 9*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.