Real-rooted atlas

A038291 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114822
219622464484
327444704268851210648
4384168780875264286724096234256
5537824153664017561601003520286720327685153632
67529536258155523687936028098560120422402752512262144113379904
71054135044216540167228354566884147203933798401348730882569011220971522494357888
814757890566746464256134929285121542048972811014635520503526195214386462722348810241677721654875873536
920661046784106256812032242872713216323830284288277568815104158610751488604231434241479750451221139292161342177281207269217792
1028925465497616528837427204250272481280647660568576064766056857604441101041664211481001984069055021056014797504512018790481920107374182426559922791424
11404956516966425454409637888727268846796801246746594508801424853250867201139882600693766513614861107226586183106560759605231616014468671078401653562408968589934592584318301411328
125669391237529638875825628774412218116626186242327260309749760299219182682112027357182416650241823812161110016893295752380416319034197278720810245580390401388992423526414431090114566871947673612855002631049216
13793714773254144589616688703078420215429326962688423561376374456326050876805349376062237589997879296474191161888604162709663782220595211612844780945408368661739077632084265540360601613132292004249612506944765952549755813888282810057883082752
141111200682555801688896054604464128330185345673723904754709361539940352118597185384847769613553964043982602241161768346627080192758705859021766656379352929510883328144515401718431744412901147766947848579764109443072122568058706329610775213952204843980465111046221821273427820544
Technical details
Node id
n-a70ed65b45a0249a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-d36c4f1a2e9bd6b7

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:          14,            8;
  2:         196,          224,           64;
  3:        2744,         4704,         2688,          512;
  4:       38416,        87808,        75264,        28672,         4096;
  5:      537824,      1536640,      1756160,      1003520,       286720,        32768;
  6:     7529536,     25815552,     36879360,     28098560,     12042240,      2752512,      262144;
  7:   105413504,    421654016,    722835456,    688414720,    393379840,    134873088,    25690112,     2097152;
  8:  1475789056,   6746464256,  13492928512,  15420489728,  11014635520,   5035261952,  1438646272,   234881024,   16777216;
  9: 20661046784, 106256812032, 242872713216, 323830284288, 277568815104, 158610751488, 60423143424, 14797504512, 2113929216, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (14+8*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[(14 + 8*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] = (14 + 8*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.