Real-rooted atlas

A013622 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(8 + 5\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18513
26410525194
351216859751253297
440962446525975775062562911
5327683385255898753081255625031251328668
62621444564665123103751003625031156253843751562530689059
720971526065136524441007529238562513741875028262500251562578125767819217
81677721679895350547026381757946919750530965312516197562502369062501593750039062520647932396
9134217728104702816458864225467520629234512518840488625080000221875171012187501869843750984375001953125593015660423
1010737418241367847500251647911992375518793521125063067985406253582031753125104984742187516590937500014074218750595703125976562518092972473599
11858993459217835704594453034633961287512756206982062520253003519375014974104055625057804360796875123833402343751505109375000101943359375354492187548828125583769993093162
1268719476736232293656457455551519653289753085659303938750630865133452812559541045143250002945270913062500821505434062500134161951171875129285449218757153222656252080078125024414062519841468413649581
1354975581388830232535077836510108882204150275737459238172361251920705338864812502280287056453656251416908172680000005005108823000000010547300826562500135602263671875010613141601562548870605468751206054687501220703125708013289923841153
1443980465111043932978339188185183471506428596775174670065881718225057467045679076556258485299955729471875652439458441082812528606508802300000007565258808250000001246057038789062501293573531250000083854189453125032636474609375692138671875610351562526446098726537168314

Computed in your browser ?

Technical details
Node id
n-034190847bef139d · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-1cfe5e80b35ba55f

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): no

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,           5;
  2:        64,         105,          25;
  3:       512,        1685,         975,          125;
  4:      4096,       24465,       25975,         7750,          625;
  5:     32768,      338525,      589875,       308125,        56250,        3125;
  6:    262144,     4564665,    12310375,     10036250,      3115625,      384375,       15625;
  7:   2097152,    60651365,   244410075,    292385625,    137418750,    28262500,     2515625,      78125;
  8:  16777216,   798953505,  4702638175,   7946919750,   5309653125,  1619756250,   236906250,   15937500,   390625;
  9: 134217728, 10470281645, 88642254675, 206292345125, 188404886250, 80000221875, 17101218750, 1869843750, 98437500, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (8+5*x)*P(n-1,x) + 5*x*d/dx 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[(8 + 5*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], 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] = (8 + 5*x)*v[i - 1] + 5*x*deriv(v[i - 1], x)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-f7e2151b905e00c8
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-f7e2151b905e00c8
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-f7e2151b905e00c8

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [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
  4. [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
  5. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741