Real-rooted atlas

A038303 → 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
\(P_{n}(x) = \left(11 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111314
2121669196
313311089297272744
41464115972653411888138416
5161051219615119790326704455243537824
6177156128989181976535718740147015160387297529536
7194871713720278130438639138357453773385617463561332187105413504
82143588814676921044464333722435091128301447018112248246985219245665611475789056
9235794769157876897876313843404401790034816436865064482781388150511695265726495391968320661046784
1025937424601707384307308681534680563138434040301342526109862119036224139069034930764035724645216513059049289254654976
1128531167061185593501183311671841070459549688148555208920808301988860672265424165469810566556110144089401513099036571449291771474049565169664
1231384283767211027122014199615406830212994140062092845408594719333695375042298197611933164033562789570813044754950249557635760604715653142338340453144156693912375296
1334522712143931122398706692119200288792768922200288792768922136560540524265670388108028212437774938284466484771044121359915771357206047844145224779466341671913386759960631594323793714773254144
1437974983358324114499539100451022570372840534499280404309876490821030323240736811147108540403826469271675619747146266496297064349045047981636346273599666865400945409858248871485851696851245525742478296911112006825558016

Generated b-files: A001023 (row sums), A038229 (reversed triangle), A038317 ?

Computed in your browser ?

Technical details
Node id
n-3c11bfaa4e68c9bc · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-4f0d71d8efeb5cf0

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:         11,          3;
  2:        121,         66,          9;
  3:       1331,       1089,        297,         27;
  4:      14641,      15972,       6534,       1188,         81;
  5:     161051,     219615,     119790,      32670,       4455,       243;
  6:    1771561,    2898918,    1976535,     718740,     147015,     16038,      729;
  7:   19487171,   37202781,   30438639,   13835745,    3773385,    617463,    56133,    2187;
  8:  214358881,  467692104,  446433372,  243509112,   83014470,  18112248,  2469852,  192456,   6561;
  9: 2357947691, 5787689787, 6313843404, 4017900348, 1643686506, 448278138, 81505116, 9526572, 649539, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (11+3*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[(11 + 3*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] = (11 + 3*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.

A038229: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,     11;
  2:     9,     66,     121;
  3:    27,    297,    1089,     1331;
  4:    81,   1188,    6534,    15972,     14641;
  5:   243,   4455,   32670,   119790,    219615,     161051;
  6:   729,  16038,  147015,   718740,   1976535,    2898918,    1771561;
  7:  2187,  56133,  617463,  3773385,  13835745,   30438639,   37202781,   19487171;
  8:  6561, 192456, 2469852, 18112248,  83014470,  243509112,  446433372,  467692104,  214358881;
  9: 19683, 649539, 9526572, 81505116, 448278138, 1643686506, 4017900348, 6313843404, 5787689787, 2357947691;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+3*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[(11 + 3*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] = (11 + 3*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.

A038317: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,          3;
  2:        121,         66,          9;
  3:       1331,       1089,        297,         27;
  4:      14641,      15972,       6534,       1188,         81;
  5:     161051,     219615,     119790,      32670,       4455,       243;
  6:    1771561,    2898918,    1976535,     718740,     147015,     16038,      729;
  7:   19487171,   37202781,   30438639,   13835745,    3773385,    617463,    56133,    2187;
  8:  214358881,  467692104,  446433372,  243509112,   83014470,  18112248,  2469852,  192456,   6561;
  9: 2357947691, 5787689787, 6313843404, 4017900348, 1643686506, 448278138, 81505116, 9526572, 649539, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (11+3*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[(11 + 3*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] = (11 + 3*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-254f1b828fe6620e
  • 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-254f1b828fe6620e
  • 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-254f1b828fe6620e

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 (2)

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