Real-rooted atlas

A038259 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116521
225616025441
34096384012001259261
46553681920384008000625194481
51048576163840010240003200005000031254084101
61677721631457280245760001024000024000003000001562585766121
726843545658720256055050240028672000089600000168000001750000781251801088541
842949672961073741824011744051200734003200028672000007168000001120000001000000039062537822859361
96871947673619327352832024159191040017616076800082575360000258048000005376000000720000000562500001953125794280046581
1010995116277763435973836800483183820800040265318400002202009600000825753600000215040000000384000000004500000000312500000976562516679880978201
111759218604441660473139527680944892805120008858370048000055364812800000242221056000007569408000000168960000000026400000000027500000000171875000048828125350277500542221
1228147497671065610555311626649601814194185830400188978561024000013287555072000006643777536000002422210560000006488064000000012672000000000176000000000016500000000093750000002441406257355827511386641
1345035996273704961829587348619264034304762786611200393075406929920003070901616640000017273821593600000719742566400000022491955200000005271552000000009152000000000011440000000000975000000000507812500001220703125154472377739119461
14720575940379279363152519739159347206403555720167424008004444650209280006878819621273600004299262263296000002015279185920000007197425664000000019680460800000000410009600000000064064000000000072800000000000568750000000027343750000061035156253243919932521508681
Technical details
Node id
n-bd93be7e16f9aab7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-833b492313a93e6e

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:          16,            5;
  2:         256,          160,           25;
  3:        4096,         3840,         1200,          125;
  4:       65536,        81920,        38400,         8000,         625;
  5:     1048576,      1638400,      1024000,       320000,       50000,        3125;
  6:    16777216,     31457280,     24576000,     10240000,     2400000,      300000,      15625;
  7:   268435456,    587202560,    550502400,    286720000,    89600000,    16800000,    1750000,     78125;
  8:  4294967296,  10737418240,  11744051200,   7340032000,  2867200000,   716800000,  112000000,  10000000,   390625;
  9: 68719476736, 193273528320, 241591910400, 176160768000, 82575360000, 25804800000, 5376000000, 720000000, 56250000, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (16+5*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[(16 + 5*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] = (16 + 5*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.