Real-rooted atlas

A038220 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(25 + 60\,x + 36\,x^{2} + 30\,n + 36\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111617
28513236253
3575153011882163509
4362513800183609504129646585
521875108750207000183600712807776600281
6128125787500195750024840001652400513216466567569397
7734375538125016537500274050002608200013880160359251227993693892733

All 15 rows

Technical details
Node id
n-1aa95e648a4ec383 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-98c8469350411ee3

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:       11,         6;
  2:       85,       132,        36;
  3:      575,      1530,      1188,        216;
  4:     3625,     13800,     18360,       9504,       1296;
  5:    21875,    108750,    207000,     183600,      71280,       7776;
  6:   128125,    787500,   1957500,    2484000,    1652400,     513216,      46656;
  7:   734375,   5381250,  16537500,   27405000,   26082000,   13880160,    3592512,    279936;
  8:  4140625,  35250000, 129150000,  264600000,  328860000,  250387200,  111041280,  24634368,   1679616;
  9: 23046875, 223593750, 951750000, 2324700000, 3572100000, 3551688000, 2253484800, 856604160, 166281984, 10077696;
%F Formula
The row polynomials satisfy (6*n-1+6*x)*P(n,x) = (30*n+25+(36*n+60)*x+36*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[((30*n + 25 + (36*n + 60)*x + 36*x^2)*P[n - 1, x])/(6*n - 1 + 6*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] = ((30*n + 25 + (36*n + 60)*x + 36*x^2)*v[i - 1])/(6*n - 1 + 6*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.