Real-rooted atlas

A038255 → 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
\(P_{n}(x) = \left(7 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1729
24928481
3343294848729
4240127441176224166561
516807240101372039205603259049
61176492016841440605488011760134464531441
7823543164708614117886722801920803292831361284782969

Show all 15 stored rows

Generated b-files: A001019 (row sums), A013623 (reversed triangle), A038268, A100062 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-6a4f4798749d2440 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-8d10fbbacee008e8

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,         2;
  2:       49,        28,         4;
  3:      343,       294,        84,        8;
  4:     2401,      2744,      1176,      224,       16;
  5:    16807,     24010,     13720,     3920,      560,      32;
  6:   117649,    201684,    144060,    54880,    11760,    1344,      64;
  7:   823543,   1647086,   1411788,   672280,   192080,   32928,    3136,    128;
  8:  5764801,  13176688,  13176688,  7529536,  2689120,  614656,   87808,   7168,   256;
  9: 40353607, 103766418, 118590192, 79060128, 33882912, 9680832, 1843968, 225792, 16128, 512;
%F Formula
The row polynomials satisfy P(n,x) = (7+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[(7 + 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] = (7 + 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.

A013623: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,     7;
  2:   4,    28,     49;
  3:   8,    84,    294,     343;
  4:  16,   224,   1176,    2744,    2401;
  5:  32,   560,   3920,   13720,   24010,    16807;
  6:  64,  1344,  11760,   54880,  144060,   201684,   117649;
  7: 128,  3136,  32928,  192080,  672280,  1411788,  1647086,    823543;
  8: 256,  7168,  87808,  614656, 2689120,  7529536, 13176688,  13176688,   5764801;
  9: 512, 16128, 225792, 1843968, 9680832, 33882912, 79060128, 118590192, 103766418, 40353607;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+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[(7 + 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] = (7 + 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.

A038268: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        7,         2;
  2:       49,        28,         4;
  3:      343,       294,        84,        8;
  4:     2401,      2744,      1176,      224,       16;
  5:    16807,     24010,     13720,     3920,      560,      32;
  6:   117649,    201684,    144060,    54880,    11760,    1344,      64;
  7:   823543,   1647086,   1411788,   672280,   192080,   32928,    3136,    128;
  8:  5764801,  13176688,  13176688,  7529536,  2689120,  614656,   87808,   7168,   256;
  9: 40353607, 103766418, 118590192, 79060128, 33882912, 9680832, 1843968, 225792, 16128, 512;
%F Formula
The row polynomials satisfy P(n,x) = (7+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[(7 + 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] = (7 + 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-4c464713c83c5f6b
  • 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-4c464713c83c5f6b
  • 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-4c464713c83c5f6b

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