Real-rooted atlas

A038255 → 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
\(P_{n}(x) = \left(7 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
249429100
3343441189271000
42401411626467568110000
5168073601530870132302835243100000
611764930252632413518522059535102067291000000
782354324706293176523226894597240525004735721218710000000
8576480119765032296475482541218413613670466754410001881224726561100000000
94035360715564962726682793226682793217153224273513818210039483857868413343196831000000000
1028247524912106082102334744405266827932020012094901029193452367569090900169201446700513778105904910000000000
11197732674393216832171997503546525682188455220133043901320798263456605639861732825710371319795530456854546773177147100000000000
121384128720171183762748167790297906239700425580231139696095158495791608792478958042911147192877977156951485279180190964466148803485314411000000000000
13968890104075398102008391388083373586218127387277823370791494051802889629541103022264545244152399090814191842564933790101345579258880267705583448361131159432310000000000000
1467822307284940693384370941133601421761919433167230204229033756641691963146485500212620227406787618133587271223180009522676622859863621419184256492211715724423696954191562436544782969100000000000000

Generated b-files: A011557 (row sums), A013624 (reversed triangle), A038269 ?

Computed in your browser ?

Technical details
Node id
n-67dfd62e07739d13 · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-5184fe74b2189306

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:        7,         3;
  2:       49,        42,         9;
  3:      343,       441,       189,        27;
  4:     2401,      4116,      2646,       756,        81;
  5:    16807,     36015,     30870,     13230,      2835,      243;
  6:   117649,    302526,    324135,    185220,     59535,    10206,      729;
  7:   823543,   2470629,   3176523,   2268945,    972405,   250047,    35721,    2187;
  8:  5764801,  19765032,  29647548,  25412184,  13613670,  4667544,  1000188,  122472,   6561;
  9: 40353607, 155649627, 266827932, 266827932, 171532242, 73513818, 21003948, 3857868, 413343, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (7+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[(7 + 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] = (7 + 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.

A013624: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,      7;
  2:     9,     42,      49;
  3:    27,    189,     441,      343;
  4:    81,    756,    2646,     4116,     2401;
  5:   243,   2835,   13230,    30870,    36015,     16807;
  6:   729,  10206,   59535,   185220,   324135,    302526,    117649;
  7:  2187,  35721,  250047,   972405,  2268945,   3176523,   2470629,    823543;
  8:  6561, 122472, 1000188,  4667544, 13613670,  25412184,  29647548,  19765032,   5764801;
  9: 19683, 413343, 3857868, 21003948, 73513818, 171532242, 266827932, 266827932, 155649627, 40353607;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+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[(7 + 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] = (7 + 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.

A038269: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        7,         3;
  2:       49,        42,         9;
  3:      343,       441,       189,        27;
  4:     2401,      4116,      2646,       756,        81;
  5:    16807,     36015,     30870,     13230,      2835,      243;
  6:   117649,    302526,    324135,    185220,     59535,    10206,      729;
  7:   823543,   2470629,   3176523,   2268945,    972405,   250047,    35721,    2187;
  8:  5764801,  19765032,  29647548,  25412184,  13613670,  4667544,  1000188,  122472,   6561;
  9: 40353607, 155649627, 266827932, 266827932, 171532242, 73513818, 21003948, 3857868, 413343, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (7+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[(7 + 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] = (7 + 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-b9cdb58d54bdc029
  • 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-b9cdb58d54bdc029
  • 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-b9cdb58d54bdc029

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