Real-rooted atlas

A027466 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + n x\right)\,P_{n}(x) = \left(32 + 16\,x + 2\,x^{2} + 8\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
2646412140
3512768288321600
440968192460810248018000
5327688192061440204803200192200000
62621447864327372803276807680092164482200000
72097152734003282575364587520143360025804825088102424000000
816777216671088648808038458720256229376005505024802816655362304260000000
91342177286039797769059696647046430723303014409909043219267584235929616588851202800000000
101073741824536870912090596966408053063680440401920015854469123853516806291456066355204096001126430000000000
1185899345924724464025688583700480885837004805536481280023253221376678218956813841203201946419201802240099123224576320000000000
1268719476736412316860416850403524608944892805120664377753600318901321728108515033088265751101444671406080576716800475791362359296532483400000000000
1354975581388835734127902728040178778112982688517324876772540416004145717182464161222334873646063524249697165246464149946368001649410048122683392553779211468836000000000000
14439804651110430786325577728750416685957121000555581276168598524526592051591147159552225711268823047370163879936181375126732833587986432046183481344458017996831011635212845056245760380000000000000
Technical details
Node id
n-7af05cad24952808 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-af50e1602aec03e4

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:         8,         4;
  2:        64,        64,        12;
  3:       512,       768,       288,        32;
  4:      4096,      8192,      4608,      1024,        80;
  5:     32768,     81920,     61440,     20480,      3200,      192;
  6:    262144,    786432,    737280,    327680,     76800,     9216,      448;
  7:   2097152,   7340032,   8257536,   4587520,   1433600,   258048,    25088,    1024;
  8:  16777216,  67108864,  88080384,  58720256,  22937600,  5505024,   802816,   65536,   2304;
  9: 134217728, 603979776, 905969664, 704643072, 330301440, 99090432, 19267584, 2359296, 165888, 5120;
%F Formula
The row polynomials satisfy (4+n*x)*P(n,x) = (32+(8*n+16)*x+(2*n+2)*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[((32 + (8*n + 16)*x + (2*n + 2)*x^2)*P[n - 1, x])/(4 + n*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] = ((32 + (8*n + 16)*x + (2*n + 2)*x^2)*v[i - 1])/(4 + n*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.