Real-rooted atlas

A013610 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 9\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
199
272162234
3432194421874563
4230415552349922624479092
5115201036803499205248802952451285245
655296622080279936062985607085880318864620049822
725804834836481959552058786560992023208928208833480783304088967
81179648185794561254113284702924801058158080142851340810713850563443737684517893224
953084169555148875246796833861058569523422720171421608961928493100812397455648348678440166074188401
10235929604777574404299816960225740390407618738176017142160896025713241344024794911296013947137604034867844010954404943570
111038090242335703040236489932801418939596805587074662401508510158848282845654784036365869900803068370272880153418513644034519165569913647990693051
124529848321121137459212612796416085136375808038311369113601206808127078427153182859264436390438809604909392436608036820443274560165691994735523389154437772193553322555996
1319629342725299922534465586541363249189906022402490238992384089648603725824235327584780288453846056361984638221016759040638221016759040430799186312352176236030764144330442557682772725875959330277
148455716864247329718272333895119667227546347372544154948203970560627540226080768188262067824230442358965260451847148075387701248893509423462656080415848111639044934608861396032185047832302351232027509436945438162263430623878
Technical details
Node id
n-c7e6b62134d63fdb · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-12e1fe8c90bb4450

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:      72,      162;
  3:     432,     1944,      2187;
  4:    2304,    15552,     34992,      26244;
  5:   11520,   103680,    349920,     524880,     295245;
  6:   55296,   622080,   2799360,    6298560,    7085880,     3188646;
  7:  258048,  3483648,  19595520,   58786560,   99202320,    89282088,    33480783;
  8: 1179648, 18579456, 125411328,  470292480, 1058158080,  1428513408,  1071385056,   344373768;
  9: 5308416, 95551488, 752467968, 3386105856, 9523422720, 17142160896, 19284931008, 12397455648, 3486784401;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*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[((4*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] = ((4*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.