Real-rooted atlas

A038243 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 3\,x + 9\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(54 + 54\,x + 18\,x^{2} + 2\,x^{3} + 54\,n x + 30\,n x^{2} + 4\,n x^{3} + 6\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16814
2369636168
32168646481281856
4129669127776307240019456
57776518407776046080120001152196608
6466563732486998405529602160004147231361933312
72799362612736587865658060803024000870912131712819218612224
816796161791590447029248557383683628800013934592316108839321620736176160768
91007769612093235236279705650164531239191040018811699256899584106168321119744512001644167168
1060466176806215680272097792042998169603919104000225740390485349376021233664033592320307200012390415166603264
113627970565321023488199538380803547348992036951552000248314429441126611763235035545607390310401013760008177664294912138512695296
1221767823363482851737614366763417628378791936033256396800025540912742413519341158450451185664133025587202433024000294395904212336646922241254130450432
13130606940162263853629441018734133248221354577100828822210560002490238992384150644087193665586541363220751991603247443968000765429350482811289653993472160563211269994184704
14783641640961462797729792713113893273616903440433152242106568704002324223059558415817629155328787038496358429052788244487970586624001607401635842318716108822677258241348730883686400100605313941504
Technical details
Node id
n-51a12c014558e493 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1e916cb7d7a3299c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,         8;
  2:       36,        96,        36;
  3:      216,       864,       648,       128;
  4:     1296,      6912,      7776,      3072,       400;
  5:     7776,     51840,     77760,     46080,     12000,      1152;
  6:    46656,    373248,    699840,    552960,    216000,     41472,     3136;
  7:   279936,   2612736,   5878656,   5806080,   3024000,    870912,   131712,     8192;
  8:  1679616,  17915904,  47029248,  55738368,  36288000,  13934592,  3161088,   393216,   20736;
  9: 10077696, 120932352, 362797056, 501645312, 391910400, 188116992, 56899584, 10616832, 1119744, 51200;
%F Formula
The row polynomials satisfy (9+(9*n-3)*x+n^2*x^2)*P(n,x) = (54+(54*n+54)*x+(6*n^2+30*n+18)*x^2+(2*n^2+4*n+2)*x^3)*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[((54 + (54*n + 54)*x + (6*n^2 + 30*n + 18)*x^2 + (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x])/(9 + (9*n - 3)*x + n^2*x^2)]];
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] = ((54 + (54*n + 54)*x + (6*n^2 + 30*n + 18)*x^2 + (2*n^2 + 4*n + 2)*x^3)*v[i - 1])/(9 + (9*n - 3)*x + n^2*x^2)); 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.