Real-rooted atlas

A038327 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(17 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
117421
228913616441
349133468816649261
48352178608277444352256194481
5141985716704207860801849602176010244084101
624137569340765682004504062886401109760104448409685766121
7410338673675851932477071952187087040440204806214656487424163841801088541
869757574411313083753610813630912508876748814966963202817310723314483222282246553637822859361
91185878764972511272678762363550756481297635709444579890739210776213504169038643217045913610027008262144794280046581
102015993900449474351505988050225453575203151401008640129763570944036639125913671841423360965935104085229568044564480104857616679880978201
11342718963076338870373161975610435733131736073663998576960346654110950401141919424307226868692336644515746611205312643072041667788801960837124194304350277500542221
125826222372297611645051022766384212888955887414416697173010777608839679829235203327879465123849135355394457618424246173696270944796672028334096384020000538624855638016167772167355827511386641
139904578032905937302963563359475724277132659192598436900752353818496217063249140108809193267022404608288416220310732867862640073113611975760012902415654588252160147337301196894548000768370776473667108864154472377739119461
1416837782655940092955465636984273247284829797740653201679839809638261836851661053295345894424311083903692185685803825542443008230732976248586244750384805117952745158400802816876656942120967500808060928441224003584159719096322684354563243919932521508681
Technical details
Node id
n-41b6e8702634279f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2fc5396437dfa965

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,            4;
  2:          289,          136,           16;
  3:         4913,         3468,          816,           64;
  4:        83521,        78608,        27744,         4352,         256;
  5:      1419857,      1670420,       786080,       184960,       21760,        1024;
  6:     24137569,     34076568,     20045040,      6288640,     1109760,      104448,       4096;
  7:    410338673,    675851932,    477071952,    187087040,    44020480,     6214656,     487424,     16384;
  8:   6975757441,  13130837536,  10813630912,   5088767488,  1496696320,   281731072,   33144832,   2228224,    65536;
  9: 118587876497, 251127267876, 236355075648, 129763570944, 45798907392, 10776213504, 1690386432, 170459136, 10027008, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (17+4*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 + 4*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 + 4*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

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.