Real-rooted atlas

A038277 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 11\,n x\right)\,P_{n}(x) = \left(162 + 396\,x + 242\,x^{2} + 198\,n x + 242\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1184462
2324158414523360
35832427687840842592169600
410497610264322822688306662411712808192000
51889568230947208468064013799808010541520030921792384000000
634012224498845952228637728049679308805692420800333955353679365932817612800000
7612220032104757649925761670745615648982272023908167360021039187276810000107532819954863104794624000000

All 15 rows

Technical details
Node id
n-b6e9da715b20f783 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2a7e84a0ad2b08f0

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:           18,            44;
  2:          324,          1584,           1452;
  3:         5832,         42768,          78408,           42592;
  4:       104976,       1026432,        2822688,         3066624,         1171280;
  5:      1889568,      23094720,       84680640,       137998080,       105415200,        30921792;
  6:     34012224,     498845952,     2286377280,      4967930880,      5692420800,      3339553536,       793659328;
  7:    612220032,   10475764992,    57616707456,    156489822720,    239081673600,    210391872768,    100001075328,     19954863104;
  8:  11019960576,  215501451264,  1382800978944,   4506906894336,   8606940249600,  10098809892864,   7200077423616,   2873500286976,   493882861824;
  9: 198359290368, 4363904388096, 32001965512704, 121686486147072, 278864864087040, 409001800660992, 388804180875264, 232753523245056, 80009023615488, 12072692177920;
%F Formula
The row polynomials satisfy (9+11*n*x)*P(n,x) = (162+(198*n+396)*x+(242*n+242)*x^2)*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[Together[((162 + (198*n + 396)*x + (242*n + 242)*x^2)*P[n - 1, x])/(9 + 11*n*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] = ((162 + (198*n + 396)*x + (242*n + 242)*x^2)*v[i - 1])/(9 + 11*n*x)); 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.