Real-rooted atlas

A038279 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(9 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19211
281364121
372948610881331
46561583219442881614641
5590496561029160648072032161051
6531441708588393660116640194401728641771561
7478296974401744960116183708040824054432403212819487171
843046721765275045952139226453952734832013063681451529216256214358881
9387420489774840978688747536357128352119042784264539523919104373248207365122357947691
1034867844017748409780774840978045916502401785641760476171136881798401119744093312046080102425937424601
113138105960976709256822852325075805682167172025254076320785682374417459608322771366403079296022809601013762048285311670611
1228242953648175314543061692051108186468186006064034093003032012121956633631427294976598615142483140992082114560547430422118440963138428376721
1325418658283297343167948506979089059800879777627094884432090394160177283615766452528478745611672995276819454992128240185088021349785612939264479232819234522712143931
1422876792454961711722431932121028043512790849138164558140855844338966416248197062072968273235402432210113914982440855483468860526642176672518246454344908830191616103219216384379749833583241

Generated b-files: A001020 (row sums), A038215 (reversed triangle), A038292 ?

Computed in your browser ?

Technical details
Node id
n-20a6b63b1c7531be · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-0205823d5b324a40

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:         9,         2;
  2:        81,        36,         4;
  3:       729,       486,       108,         8;
  4:      6561,      5832,      1944,       288,        16;
  5:     59049,     65610,     29160,      6480,       720,       32;
  6:    531441,    708588,    393660,    116640,     19440,     1728,      64;
  7:   4782969,   7440174,   4960116,   1837080,    408240,    54432,    4032,    128;
  8:  43046721,  76527504,  59521392,  26453952,   7348320,  1306368,  145152,   9216,   256;
  9: 387420489, 774840978, 688747536, 357128352, 119042784, 26453952, 3919104, 373248, 20736, 512;
%F Formula
The row polynomials satisfy P(n,x) = (9+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[(9 + 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] = (9 + 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)

Generated with polytool c48629374597.

A038215: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,     9;
  2:   4,    36,     81;
  3:   8,   108,    486,     729;
  4:  16,   288,   1944,    5832,     6561;
  5:  32,   720,   6480,   29160,    65610,     59049;
  6:  64,  1728,  19440,  116640,   393660,    708588,    531441;
  7: 128,  4032,  54432,  408240,  1837080,   4960116,   7440174,   4782969;
  8: 256,  9216, 145152, 1306368,  7348320,  26453952,  59521392,  76527504,  43046721;
  9: 512, 20736, 373248, 3919104, 26453952, 119042784, 357128352, 688747536, 774840978, 387420489;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (9+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[(9 + 2*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] = (9 + 2*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.

A038292: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         9,         2;
  2:        81,        36,         4;
  3:       729,       486,       108,         8;
  4:      6561,      5832,      1944,       288,        16;
  5:     59049,     65610,     29160,      6480,       720,       32;
  6:    531441,    708588,    393660,    116640,     19440,     1728,      64;
  7:   4782969,   7440174,   4960116,   1837080,    408240,    54432,    4032,    128;
  8:  43046721,  76527504,  59521392,  26453952,   7348320,  1306368,  145152,   9216,   256;
  9: 387420489, 774840978, 688747536, 357128352, 119042784, 26453952, 3919104, 373248, 20736, 512;
%F Formula
The row polynomials satisfy P(n,x) = (9+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[(9 + 2*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] = (9 + 2*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.

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-0da8e8e5bf86fd8d
  • 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-0da8e8e5bf86fd8d
  • 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-0da8e8e5bf86fd8d

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