Real-rooted atlas

A130123 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(2 + 12\,x + 18\,x^{2} + 6\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18614
2289636160
3805048642161664
4208192060486912129616384
551262402880060480518407776155648
6121618432112320345600544320373248466561441792
7281651072387072157248036288004572288261273627993613107200
8640013516812257286193152188697603483648036578304179159041679616117440512
91433634560036495362206310483607552203793408313528320282175488120932352100776961040187392
103174486016010368000729907203309465601003290624203793408026873856002116316160806215680604661769126805504
11696322095104283852802280960001204346880436849459211036196864192148070402217093120015519651840532102348836279705679456894976
1215155250135047542374468124672041057280001734259507252421935104113515167744172933263360177367449600111741493248348285173762176782336687194767360
133276801182105619552665619610173441328431104064049356800225453735936584130134016110677288550414987549491201383466106880792348770304226385362944130606940165909874999296
147045122752512049648435254747463684118136422422317642547289669099520027054448312326133366407168103298802647041258954157260810564650270720554644139212814627977297927836416409650577534877696
Technical details
Node id
n-2b577edb24bfe619 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1f45c55ca90e4ac8

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,      6;
  2:    28,     96,      36;
  3:    80,    504,     864,      216;
  4:   208,   1920,    6048,     6912,     1296;
  5:   512,   6240,   28800,    60480,    51840,      7776;
  6:  1216,  18432,  112320,   345600,   544320,    373248,     46656;
  7:  2816,  51072,  387072,  1572480,  3628800,   4572288,   2612736,    279936;
  8:  6400, 135168, 1225728,  6193152, 18869760,  34836480,  36578304,  17915904,   1679616;
  9: 14336, 345600, 3649536, 22063104, 83607552, 203793408, 313528320, 282175488, 120932352, 10077696;
%F Formula
The row polynomials satisfy (3*n-2+3*x)*P(n,x) = (6*n+2+(18*n+12)*x+18*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[((6*n + 2 + (18*n + 12)*x + 18*x^2)*P[n - 1, x])/(3*n - 2 + 3*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] = ((6*n + 2 + (18*n + 12)*x + 18*x^2)*v[i - 1])/(3*n - 2 + 3*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.