Real-rooted atlas

A038271 → x ↦ x/(x+1) → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(7\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 12\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
11212
2168288456
317646048518412996
4164648467214515282944329232
51440609878402540160290304012441607819260
612101041037232035562240609638405225472017915904178279128
7988251610164873643563744099574272012802406408778792962508226563951854004

All 15 rows

Technical details
Node id
n-0ea2fa864539dca6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-48a0cbb267bcb941

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:        12;
  2:       168,        288;
  3:      1764,       6048,        5184;
  4:     16464,      84672,      145152,        82944;
  5:    144060,     987840,     2540160,      2903040,      1244160;
  6:   1210104,   10372320,    35562240,     60963840,     52254720,     17915904;
  7:   9882516,  101648736,   435637440,    995742720,   1280240640,    877879296,    250822656;
  8:  79060128,  948721536,  4879139328,  13940398080,  23897825280,  24580620288,  14046068736,   3439853568;
  9: 622598508, 8538493824, 51230962944, 175649015808, 376390748160, 516193026048, 442451165184, 216710774784, 46438023168;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (7*n+12*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 12.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 12;
P[n_, x_] := P[n, x] = Expand[Together[((7*n + 12*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] = 12; for(n = 2, N, my(i = n + 1); v[i] = ((7*n + 12*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.