Real-rooted atlas

A038291 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1141226
2196336144676
3274470566048172817576
4384161317121693449676820736456976
5537824230496039513603386880145152024883211881376
6752953638723328829785609483264060963840209018882985984308915776
71054135046324810241626379776232339968019914854401024192512292626432358318088031810176
814757890561011969638430359089152520441528325576159232038236520448163870801924013162496429981696208827064576
9206610467841593852180485464636047361092927209472140519212646412044503941126882573680642528292372485417769369651597803525429503678976
10289254654976247932561408095631130828802185854418944032787816284160337246110351362408900788224011798697738240379243855872072236924928061917364224141167095653376
114049565169664381816144568321636354905292804207769756467207213319582515208655983499018247419414427729924542498629222401946785126809605562243219456095352740904967430083706883670344486987776
1256693912375296583137384431616274907624089190478545035454054401514797112328192020774360397643776207743603976437761526279539418726481764975326003203114856202895360800963023601664124825406275584891610044825695428956661682176
13793714773254144884425033054617645484715985666048142951964526379008306325638270812160472616699046395904540133370338738176462971460290347008297624510186651648141725957231738880485917567651676161135911197107814416227302815825921069932053790722481152873203736576
14111120068255580161333440819066961927429170277658787842547144095197298688600398251010791833610292541445899288576132332675732990853121296320088812971622497224006660972871685555657523484164096238099608149321318474212864877710540815902756759509401620970668254298112128391846454886464509974703297150976
Technical details
Node id
n-9fa7dedda2858777 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-8e9037e0ccb64a62

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,           12;
  2:         196,          336,          144;
  3:        2744,         7056,         6048,          1728;
  4:       38416,       131712,       169344,         96768,         20736;
  5:      537824,      2304960,      3951360,       3386880,       1451520,        248832;
  6:     7529536,     38723328,     82978560,      94832640,      60963840,      20901888,      2985984;
  7:   105413504,    632481024,   1626379776,    2323399680,    1991485440,    1024192512,    292626432,     35831808;
  8:  1475789056,  10119696384,  30359089152,   52044152832,   55761592320,   38236520448,  16387080192,   4013162496,   429981696;
  9: 20661046784, 159385218048, 546463604736, 1092927209472, 1405192126464, 1204450394112, 688257368064, 252829237248, 54177693696, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (14+12*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 + 12*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 + 12*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.