Real-rooted atlas

A013609 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(9 + 24\,x + 16\,x^{2} + 12\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2335616105
313539633664931
45132160316817922567889
518631026021600211208960102464827
6656144712123120172800126720430084096521017
72259918370862596811491201209600709632200704163844117715
876545723168293932866769929192960774144037847049175046553632118177
925587927556201301702435271936600929286618931246448640194641924128768262144247886443
108463691023516055112400173560320352719360480743424441262080265420800973209601835008010485761896619529
112775303372402362251735208083152001909163520310393036835254517762773647360145981440047579136080740352419430414406237699
12903449713321454489376566436027763209699782400183279697922483144294424174526464166418841607785676800228379852835232153616777216108752970865
132922925546979384434635781441549193817646836092160100877736960158842404864184462147584157134422016961531084804048551936010796138496152672665667108864816635944859
149406505716368382801315422763264653458688216887134464524564232192941525544960127073923891212912350330889777252925445384574074882061080985605038197964865766686722684354566104007655641
Technical details
Node id
n-110aab533f3cf401 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-62f46345657f9dbc

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:      7,       4;
  2:     33,      56,       16;
  3:    135,     396,      336,       64;
  4:    513,    2160,     3168,     1792,      256;
  5:   1863,   10260,    21600,    21120,     8960,     1024;
  6:   6561,   44712,   123120,   172800,   126720,    43008,     4096;
  7:  22599,  183708,   625968,  1149120,  1209600,   709632,   200704,    16384;
  8:  76545,  723168,  2939328,  6676992,  9192960,  7741440,  3784704,   917504,   65536;
  9: 255879, 2755620, 13017024, 35271936, 60092928, 66189312, 46448640, 19464192, 4128768, 262144;
%F Formula
The row polynomials satisfy (4*n-1+4*x)*P(n,x) = (12*n+9+(16*n+24)*x+16*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[((12*n + 9 + (16*n + 24)*x + 16*x^2)*P[n - 1, x])/(4*n - 1 + 4*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] = ((12*n + 9 + (16*n + 24)*x + 16*x^2)*v[i - 1])/(4*n - 1 + 4*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.