Real-rooted atlas

A038327 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114216
2196564256
32744117616884096
4384162195247044481665536
5537824384160109760156801120321048576
67529536645388823049604390404704026886416777216
7105413504105413504451772161075648015366401317126272128268435456
814757890561686616064843308032240945152430259204917248351232143362564294967296
920661046784265642030081517954457650598481921084253184154893312147517449031683225651268719476736
102892546549764132209356802656420300801011969638402529924096043370127365163110404214784022579207168010241099511627776
1140495651696646363602409472454543029248019480415539205565833011201113166602241590238003216226918401159065605519360157696204817592186044416
1256693912375296971895640719367636322891366436363442339840116882493235202671599845376445266640896545224458244868075520309084160132464643440644096281474976710656
137937147732541441474041721757696126346433293516866181465058508823636237520896060778896482304115769326632961653847523328177197948928140633292808036188163130982474547281924503599627370496
1411112006825558016222240136511160322063658410460774411792333774061568463270255409561613236293011701762836348502507524630773065318457884663316485512825077763937732198420455751687305625616056321638472057594037927936
Technical details
Node id
n-8852cf7cc7f2c858 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-d2d514e02d66ef29

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,           2;
  2:         196,          56,           4;
  3:        2744,        1176,         168,          8;
  4:       38416,       21952,        4704,        448,         16;
  5:      537824,      384160,      109760,      15680,       1120,        32;
  6:     7529536,     6453888,     2304960,     439040,      47040,      2688,       64;
  7:   105413504,   105413504,    45177216,   10756480,    1536640,    131712,     6272,    128;
  8:  1475789056,  1686616064,   843308032,  240945152,   43025920,   4917248,   351232,  14336,   256;
  9: 20661046784, 26564203008, 15179544576, 5059848192, 1084253184, 154893312, 14751744, 903168, 32256, 512;
%F Formula
The row polynomials satisfy P(n,x) = (14+2*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 + 2*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 + 2*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.