Real-rooted atlas

A147716 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(17 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
117623
228920436529
349135202183621612167
48352111791262424146881296279841
514198572505630176868062424011016077766436343
624137569511148524510134021224160561816079315246656148035889
7410338673101377789810734118926314187602228536804719254455520642799363404825447
869757574411969625630424330669552171745902727577025120213939532837754035238071296167961678310985281
911858787649737669090181453179892020843795205193623185696867281831871296192545579522912454144256981248100776961801152661463
10201599390044971152725898201130072705442010635978404160656928077904027822836240648183187129601650390681602184340608017132083206046617641426511213649
113427189630763313305559742963423480399546406024861599519724017549364366864086714506283328306051198647047715576436480136157231232016018497792011307174912362797056952809757913927
125826222372297612467576534149576479000150746682456352958911374404475087913550320252710846882841610405740753999363147955186083846944018792832010892578498560115333184102474010599424217678233621914624432020321
1399045780329059374544453450392135896235484831833464124540039194137424109888269877180080698113714513849923285241009476940811594968268742144306925630643174460181496204544084962112288768817817123635248106889625613060694016504036361936467383
1416837782655940092983198455476409870819086704491646970362694593575291336992261534082307688590418461229339366253449773592003193898883942289211372328961217471668217925122864639219336294450552456811816966488015847505925724719865446431084451758087836416409611592836324538749809
Technical details
Node id
n-e3503f969f5755f9 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dde9c6f2b382eac4

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:           17,            6;
  2:          289,          204,           36;
  3:         4913,         5202,         1836,          216;
  4:        83521,       117912,        62424,        14688,         1296;
  5:      1419857,      2505630,      1768680,       624240,       110160,        7776;
  6:     24137569,     51114852,     45101340,     21224160,      5618160,      793152,       46656;
  7:    410338673,   1013777898,   1073411892,    631418760,    222853680,    47192544,     5552064,     279936;
  8:   6975757441,  19696256304,  24330669552,  17174590272,   7577025120,  2139395328,   377540352,   38071296,   1679616;
  9: 118587876497, 376690901814, 531798920208, 437952051936, 231856968672, 81831871296, 19254557952, 2912454144, 256981248, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (17+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[(17 + 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] = (17 + 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.