Real-rooted atlas

A038220 → 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
\(P_{n}(x) = \left(5 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15611
2256036121
31254505402161331
4625300054004320129614641
53125187504500054000324007776161051
615625112500337500540000486000233280466561771561
7781256562502362500472500056700004082400163296027993619487171

Show all 15 stored rows

Generated b-files: A001020 (row sums), A038248, A038259 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-c1936dc488503f26 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

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,        6;
  2:      25,       60,        36;
  3:     125,      450,       540,       216;
  4:     625,     3000,      5400,      4320,      1296;
  5:    3125,    18750,     45000,     54000,     32400,      7776;
  6:   15625,   112500,    337500,    540000,    486000,    233280,     46656;
  7:   78125,   656250,   2362500,   4725000,   5670000,   4082400,   1632960,    279936;
  8:  390625,  3750000,  15750000,  37800000,  56700000,  54432000,  32659200,  11197440,  1679616;
  9: 1953125, 21093750, 101250000, 283500000, 510300000, 612360000, 489888000, 251942400, 75582720, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (5+6*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 + 6*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 + 6*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.

A038248: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,        6;
  2:      25,       60,        36;
  3:     125,      450,       540,       216;
  4:     625,     3000,      5400,      4320,      1296;
  5:    3125,    18750,     45000,     54000,     32400,      7776;
  6:   15625,   112500,    337500,    540000,    486000,    233280,     46656;
  7:   78125,   656250,   2362500,   4725000,   5670000,   4082400,   1632960,    279936;
  8:  390625,  3750000,  15750000,  37800000,  56700000,  54432000,  32659200,  11197440,  1679616;
  9: 1953125, 21093750, 101250000, 283500000, 510300000, 612360000, 489888000, 251942400, 75582720, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (5+6*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 + 6*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 + 6*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.

A038259: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,        5;
  2:       36,       60,        25;
  3:      216,      540,       450,       125;
  4:     1296,     4320,      5400,      3000,       625;
  5:     7776,    32400,     54000,     45000,     18750,      3125;
  6:    46656,   233280,    486000,    540000,    337500,    112500,     15625;
  7:   279936,  1632960,   4082400,   5670000,   4725000,   2362500,    656250,     78125;
  8:  1679616, 11197440,  32659200,  54432000,  56700000,  37800000,  15750000,   3750000,   390625;
  9: 10077696, 75582720, 251942400, 489888000, 612360000, 510300000, 283500000, 101250000, 21093750, 1953125;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+6*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 + 6*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 + 6*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-32b2129ae50cd22c, e-e341175dbafd44bc
  • 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-32b2129ae50cd22c, e-e341175dbafd44bc
  • 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-32b2129ae50cd22c, e-e341175dbafd44bc

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (6)

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