Real-rooted atlas

A013610 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + 9\,n x\right)\,P_{n}(x) = \left(16 + 72\,x + 81\,x^{2} + 36\,n x + 81\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141822
216144243403
364864291629166760
42564608233284665632805107653
51024230401555204665606561003542941656538
64096110592933120373248078732008503056372008724876631
7163845160965225472261273607348320011904278410416243638263752366837484
8655362359296278691841672151045878656001269789696166659897612244400643874204895333623945
9262144106168321433272321003290624423263232011428107264199991877122203992115213947137604387420489076678687774
1010485764718592071663616057330892802821754880091424858112199991877120293865615360278942752080154968195600383546284111092263435419
1141943042076180483503554560315319910401773674496006704489594881759928518656323252176896040911603638403409300303200168760365008437657271530815440151087088
121677721690596966416817061888168170618880106420469760045973642936321407942814924831032208982016490939243660805454880485120040502487602016180754903347843671583974253216851407678477
1367108864392586854479498838016874487218176614873825280029882867908608104590037680128268945811177472510576813407232709134463065600702043118434944469962748704384190922366661156355861215966063028751065922530
1426843545616911433728370994577408445193492889634432934215680185937844764672732130263760896215156648941977647653835918008327942305986334720982860365808921687726379758151685345826266512368199282280940993634315188682441542099639816323167
Technical details
Node id
n-64daf3f1e2390235 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-99b3f9f39145c1ec

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:      4,       18;
  2:     16,      144,       243;
  3:     64,      864,      2916,       2916;
  4:    256,     4608,     23328,      46656,      32805;
  5:   1024,    23040,    155520,     466560,     656100,      354294;
  6:   4096,   110592,    933120,    3732480,    7873200,     8503056,     3720087;
  7:  16384,   516096,   5225472,   26127360,   73483200,   119042784,   104162436,    38263752;
  8:  65536,  2359296,  27869184,  167215104,  587865600,  1269789696,  1666598976,  1224440064,   387420489;
  9: 262144, 10616832, 143327232, 1003290624, 4232632320, 11428107264, 19999187712, 22039921152, 13947137604, 3874204890;
%F Formula
The row polynomials satisfy (4+9*n*x)*P(n,x) = (16+(36*n+72)*x+(81*n+81)*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[((16 + (36*n + 72)*x + (81*n + 81)*x^2)*P[n - 1, x])/(4 + 9*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] = ((16 + (36*n + 72)*x + (81*n + 81)*x^2)*v[i - 1])/(4 + 9*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

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.