Real-rooted atlas

A027466 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 + x + n\right)\,P_{n}(x) = \left(49 + 14\,x + x^{2} + 7\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1819
26316180
3490189241704
4377319603783216144
52881218865490063040153248
6218491172872565959800945481458752
7164708615294376050521320551715013235613932160
812353145131766886117748161347226411027440176464133554432
99223681611117830559295096183532443630312475398411602268721285212672
106860113199223681605558915251976503204588311072606247923305880028358012415919104
11508455448275461245095073024880203826892554353838010094284213311144124509080850346588120401094656
1237569208117610146537844527674705420292099520611480677513044921122018856842281910418676351078004158961171798691840
1327682574402048839970552139659524959619619923723465949323440158984976152826399576374930556370810442697695140140491410411443109011456
1420346692185473875560416280341879793864718507778314486866973303191846581056323709649443556527991526561284735768162437767731783605733112112094627905536
Technical details
Node id
n-f2f14a72ebb5099e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4856738f02dd9a97

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,         1;
  2:       63,        16,        1;
  3:      490,       189,       24,        1;
  4:     3773,      1960,      378,       32,       1;
  5:    28812,     18865,     4900,      630,      40,      1;
  6:   218491,    172872,    56595,     9800,     945,     48,     1;
  7:  1647086,   1529437,   605052,   132055,   17150,   1323,    56,    1;
  8: 12353145,  13176688,  6117748,  1613472,  264110,  27440,  1764,   64,  1;
  9: 92236816, 111178305, 59295096, 18353244, 3630312, 475398, 41160, 2268, 72, 1;
%F Formula
The row polynomials satisfy (n+6+x)*P(n,x) = (7*n+49+(n+14)*x+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[((7*n + 49 + (n + 14)*x + x^2)*P[n - 1, x])/(n + 6 + 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] = ((7*n + 49 + (n + 14)*x + x^2)*v[i - 1])/(n + 6 + 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.