Real-rooted atlas

A027467 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
117118
2289341324
349138675115832
483521196521734681104976
514198574176054913028908511889568
62413756985191421252815982604335102134012224
741033867316896298329816997292323517195560691191612220032
8697575744132827093846758519327951199258464702751288092136111019960576
91185878764976278181696914772192228202755579617890198210523646412692104041531198359290368
1020159939004491185878764970313909084845492406407605068889490357803964175394105895601300517013570467226624
11342718963076332217593290493965223332073351150999977765135411762090111515568786559739342756193081064515895187164268410079232
125826222372297614112627556915961330555974296342608933282934034529999332953249882290162230311375611245267444134289510808601907420411156831381426176
1399045780329059377574089083986893267320791199537457657425552841484790331695355897779982656770414116286841420068404182735595959717515140511822542221120822964865671168
1416837782655940092913866409246068311853018623587908251124749702559784122018009894349449237412928746994209481995953231408282325736724851197072842553714836045211788332262992381374813367582081024
Technical details
Node id
n-a6f953758a1a61c0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e2989eb4e41050f8

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,           1;
  2:          289,          34,           1;
  3:         4913,         867,          51,          1;
  4:        83521,       19652,        1734,         68,         1;
  5:      1419857,      417605,       49130,       2890,        85,        1;
  6:     24137569,     8519142,     1252815,      98260,      4335,      102,      1;
  7:    410338673,   168962983,    29816997,    2923235,    171955,     6069,    119,     1;
  8:   6975757441,  3282709384,   675851932,   79511992,   5846470,   275128,   8092,   136,   1;
  9: 118587876497, 62781816969, 14772192228, 2027555796, 178901982, 10523646, 412692, 10404, 153, 1;
%F Formula
The row polynomials satisfy P(n,x) = (17+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 + 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 + 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.