Real-rooted atlas

A013620 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(5 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15914
2259081196
312567512157292744
462545001215014580656138416
531252812510125018225016402559049537824
6156251687507593751822500246037517714705314417529536
7781259843755315625159468752870437531000725186004354782969105413504
8390625562500035437500127575000287043750413343000372008700191318760430467211475789056
91953125316406252278125009568125002583393750465010875055801305004304672100193710244538742048920661046784
109765625175781250142382812568343750002152828125046501087500697516312507174453500048427561125193710244503486784401289254654976
1148828125966796875870117187546986328125169150781250426259968750767267943750986487356250887838620625532703172375191773142055313810596094049565169664
122441406255273437500522070312503132421875001268630859375365365687500076726794375001183784827500013317579309375106540634475005753194261650188286357654028242953648156693912375296
1312207031252856445312530849609375020360742187509162333984375296859621093757124630906250012824335631250017312853102187517312853102187512465254233575061193066237550183579198712652541865828329793714773254144
14610351562515380859375017995605468751295683593750064136337890625230890816406250623405204296875128243356312500020198328619218752423799434306250218141949087562514278382122095006425271954942751779306079830302287679245496111112006825558016

Generated b-files: A001023 (row sums), A038251, A038295 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-a13c2de84911142b · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:       5,        9;
  2:      25,       90,        81;
  3:     125,      675,      1215,       729;
  4:     625,     4500,     12150,     14580,       6561;
  5:    3125,    28125,    101250,    182250,     164025,      59049;
  6:   15625,   168750,    759375,   1822500,    2460375,    1771470,     531441;
  7:   78125,   984375,   5315625,  15946875,   28704375,   31000725,   18600435,    4782969;
  8:  390625,  5625000,  35437500, 127575000,  287043750,  413343000,  372008700,  191318760,   43046721;
  9: 1953125, 31640625, 227812500, 956812500, 2583393750, 4650108750, 5580130500, 4304672100, 1937102445, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (5+9*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[(5 + 9*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] = (5 + 9*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)

Generated with polytool c48629374597.

A038251: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,        9;
  2:      25,       90,        81;
  3:     125,      675,      1215,       729;
  4:     625,     4500,     12150,     14580,       6561;
  5:    3125,    28125,    101250,    182250,     164025,      59049;
  6:   15625,   168750,    759375,   1822500,    2460375,    1771470,     531441;
  7:   78125,   984375,   5315625,  15946875,   28704375,   31000725,   18600435,    4782969;
  8:  390625,  5625000,  35437500, 127575000,  287043750,  413343000,  372008700,  191318760,   43046721;
  9: 1953125, 31640625, 227812500, 956812500, 2583393750, 4650108750, 5580130500, 4304672100, 1937102445, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (5+9*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[(5 + 9*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (5 + 9*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038295: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         9,          5;
  2:        81,         90,         25;
  3:       729,       1215,        675,        125;
  4:      6561,      14580,      12150,       4500,        625;
  5:     59049,     164025,     182250,     101250,      28125,       3125;
  6:    531441,    1771470,    2460375,    1822500,     759375,     168750,     15625;
  7:   4782969,   18600435,   31000725,   28704375,   15946875,    5315625,    984375,     78125;
  8:  43046721,  191318760,  372008700,  413343000,  287043750,  127575000,  35437500,   5625000,   390625;
  9: 387420489, 1937102445, 4304672100, 5580130500, 4650108750, 2583393750, 956812500, 227812500, 31640625, 1953125;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+9*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[(5 + 9*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (5 + 9*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-4eef4a156684ec8f
  • 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-4eef4a156684ec8f
  • 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-4eef4a156684ec8f

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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