Real-rooted atlas

A013610 → D → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2243660
3144432324900
476834565184259212000
5384023040518405184019440150000
6184321382404147206220804665601399681800000
786016774144290304058060806531840391910497977621000000
8393216412876818579456464486406967296062705664313528326718464240000000
9176947221233664111476736334430208627056640752467968564350976241864704453496322700000000
10786432010616832063700992022295347205016453120752467968075246796804837294080181398528030233088030000000000
11346030085190451203503554560140142182403678732288066217181184827714764807094697984039907676160133025587201995383808330000000000
121509949442491416576186856243208408530944025225592832052973744947279460617420885136375808063852281856031926140928095778422784130606940163600000000000
13654311424117776056329716524646448582623232016396635340803935192481792688658684313688541830840328300796641280553386442752024902389923846791560888328489451110439000000000000
14281857228854962159616494659436544272062690099210202350878720275463473725445509269474508882639042117632929689223823367747410198528046484461191168190163704872964754092621824548549148672420000000000000
Technical details
Node id
n-59c0dcb7b3846aec · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f06e51c139cfae05

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:       3;
  2:      24,       36;
  3:     144,      432,       324;
  4:     768,     3456,      5184,      2592;
  5:    3840,    23040,     51840,     51840,     19440;
  6:   18432,   138240,    414720,    622080,    466560,    139968;
  7:   86016,   774144,   2903040,   5806080,   6531840,   3919104,    979776;
  8:  393216,  4128768,  18579456,  46448640,  69672960,  62705664,  31352832,   6718464;
  9: 1769472, 21233664, 111476736, 334430208, 627056640, 752467968, 564350976, 241864704, 45349632;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+6*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 6*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 6*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.