Real-rooted atlas

A038327 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115621
222518036441
33375405016202169261
4506258100048600129601296194481
5759375151875012150004860009720077764084101
61139062527337500273375001458000043740006998404665685766121
71708593754784062505740875003827250001530900003674160048988802799361801088541
8256289062582012500001148175000091854000004592700000146966400029393280033592320167961637822859361
9384433593751383960937502214337500002066715000001240029000004960116000013226976000226748160022674816010077696794280046581
10576650390625230660156250041518828125004428675000000310007250000014880348000004960116000001133740800001700611200015116544006046617616679880978201
11864975585937538058925781250761178515625009134142187500073073137500000409209570000001636838280000046766808000009353361600001247114880009976919040362797056350277500542221
1212974633789062562278242187500013701213281250001826828437500000164414559375000010522531800000004910514840000001683605088000004209012720000074826892800008979227136006530347008021767823367355827511386641
1319461950683593751012021435546875024288514453125000356231545312500003562315453125000025648671262500000136792913400000005471716536000000164151496080000036478110240000058364976384000636708833280042447255552013060694016154472377739119461
142919292602539062516348038574218750042504900292968750068007840468750000074808624515625000059846899612500000035908139767500000016415149608000000057453023628000000153208063008000003064161260160000445696183296000445696183296002742745743360783641640963243919932521508681
Technical details
Node id
n-0e639bc7a476db4f · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-9629d1e9c04dbfbe

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:          15,            6;
  2:         225,          180,           36;
  3:        3375,         4050,         1620,          216;
  4:       50625,        81000,        48600,        12960,         1296;
  5:      759375,      1518750,      1215000,       486000,        97200,        7776;
  6:    11390625,     27337500,     27337500,     14580000,      4374000,      699840,       46656;
  7:   170859375,    478406250,    574087500,    382725000,    153090000,    36741600,     4898880,     279936;
  8:  2562890625,   8201250000,  11481750000,   9185400000,   4592700000,  1469664000,   293932800,   33592320,   1679616;
  9: 38443359375, 138396093750, 221433750000, 206671500000, 124002900000, 49601160000, 13226976000, 2267481600, 226748160, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (15+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[(15 + 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] = (15 + 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.