Real-rooted atlas

A038335 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(12\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 9\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
199
2216162378
338885832218711907
46220813996810497626244333396
59331202799360314928015746402952458751645
613436928503884807558272056687040212576403188646220541454
718811699284652646415872371201587237120892820880267846264334807835403265623

All 15 rows

Technical details
Node id
n-cdaf8ce98c057e18 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c91d5d506c107735

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:         216,          162;
  3:        3888,         5832,         2187;
  4:       62208,       139968,       104976,        26244;
  5:      933120,      2799360,      3149280,      1574640,       295245;
  6:    13436928,     50388480,     75582720,     56687040,     21257640,      3188646;
  7:   188116992,    846526464,   1587237120,   1587237120,    892820880,    267846264,     33480783;
  8:  2579890176,  13544423424,  30474952704,  38093690880,  28570268160,  12856620672,   3214155168,   344373768;
  9: 34828517376, 208971104256, 548549148672, 822823723008, 771397240320, 462838344192, 173564379072, 37192366944, 3486784401;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (12*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[((12*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] = ((12*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.