Real-rooted atlas

A027466 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(11 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111617
212113236289
31331217811882164913
41464131944261369504129683521
51610514392304791602613607128077761419857
6177156157978367906140574992023522405132164665624137569
7194871717440556212175455611068596060374160197588163592512279936410338673
82143588819353842081785733488194807289613282315205795919361580705282463436816796166975757441
9235794769111575379574252553736163214320278426298984096143449004165216327424121940121616628198410077696118587876497
10259374246011414768614603472613872205051074723204821480417603155878091521434490041604471137792091455091201108546560604661762015993900449
11285311670611171187002366646687364281807639750518840833427329328063643541512323471465900672135251918208036886886784067067066880731640729636279705634271896307633
12313842837672120542440283992616273208519761120496742763201375155093391201200135354232327637224981478435706506406912121726726387202950950942720482882881536478892113922176782336582622237229761
13345227121439312447974133842388011551710756881602310342151376218496864838824021452419456902721560175960502016851005069364736348138437467392105496496202240230174173532163424078614528311279874048130606940169904578032905937
1437974983358324128999078200902041028149136213799622432344790119264336485171851788963670747329292243230033387239663808187221115260241928935553228329728324929208302899288617056809881617576936887910423968550301696201134687846478364164096168377826559400929
Technical details
Node id
n-ebafadb809217503 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-57ca34e86b523c55

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:         11,           6;
  2:        121,         132,          36;
  3:       1331,        2178,        1188,         216;
  4:      14641,       31944,       26136,        9504,        1296;
  5:     161051,      439230,      479160,      261360,       71280,        7776;
  6:    1771561,     5797836,     7906140,     5749920,     2352240,      513216,      46656;
  7:   19487171,    74405562,   121754556,   110685960,    60374160,    19758816,    3592512,     279936;
  8:  214358881,   935384208,  1785733488,  1948072896,  1328231520,   579591936,  158070528,   24634368,   1679616;
  9: 2357947691, 11575379574, 25255373616, 32143202784, 26298984096, 14344900416, 5216327424, 1219401216, 166281984, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (11+6*x)*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[(11 + 6*x)*P[n - 1, 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] = (11 + 6*x)*v[i - 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.