Real-rooted atlas

A038221 → basis A013609

palindromic γ-positive nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16612
2367236144
32166486482161728
41296518477765184129620736
57776388807776077760388807776248832
646656279936699840933120699840279936466562985984
727993619595525878656979776097977605878656195955227993635831808
816796161343692847029248940584961175731209405849647029248134369281679616429981696
910077696906992643627970568465264641269789696126978969684652646436279705690699264100776965159780352
106046617660466176027209779207255941120126978969601523747635212697896960725594112027209779206046617606046617661917364224
113627970563990767616199538380805986151424011972302848016761223987216761223987211972302848059861514240199538380803990767616362797056743008370688
12217678233626121388032143667634176478892113920107750725632017240116101122011346878464172401161011210775072563204788921139201436676341762612138803221767823368916100448256
13130606940161697890222081018734133248373535848857693383962214401680911319859222412150931456224121509314561680911319859293383962214403735358488576101873413324816978902220813060694016106993205379072
1478364164096109709829734471311389327362852455573094478442528260096156885056520192235327584780288268945811177472235327584780288156885056520192784425282600962852455573094471311389327361097098297344783641640961283918464548864

Generated b-files: A001021 (row sums), A038260, A038260 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-0416b5b5a5f283f7 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:        6,        6;
  2:       36,       72,        36;
  3:      216,      648,       648,       216;
  4:     1296,     5184,      7776,      5184,       1296;
  5:     7776,    38880,     77760,     77760,      38880,       7776;
  6:    46656,   279936,    699840,    933120,     699840,     279936,     46656;
  7:   279936,  1959552,   5878656,   9797760,    9797760,    5878656,   1959552,    279936;
  8:  1679616, 13436928,  47029248,  94058496,  117573120,   94058496,  47029248,  13436928,  1679616;
  9: 10077696, 90699264, 362797056, 846526464, 1269789696, 1269789696, 846526464, 362797056, 90699264, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (6+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[(6 + 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] = (6 + 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.

A038260: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,        6;
  2:       36,       72,        36;
  3:      216,      648,       648,       216;
  4:     1296,     5184,      7776,      5184,       1296;
  5:     7776,    38880,     77760,     77760,      38880,       7776;
  6:    46656,   279936,    699840,    933120,     699840,     279936,     46656;
  7:   279936,  1959552,   5878656,   9797760,    9797760,    5878656,   1959552,    279936;
  8:  1679616, 13436928,  47029248,  94058496,  117573120,   94058496,  47029248,  13436928,  1679616;
  9: 10077696, 90699264, 362797056, 846526464, 1269789696, 1269789696, 846526464, 362797056, 90699264, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (6+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[(6 + 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] = (6 + 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.

A038260: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,        6;
  2:       36,       72,        36;
  3:      216,      648,       648,       216;
  4:     1296,     5184,      7776,      5184,       1296;
  5:     7776,    38880,     77760,     77760,      38880,       7776;
  6:    46656,   279936,    699840,    933120,     699840,     279936,     46656;
  7:   279936,  1959552,   5878656,   9797760,    9797760,    5878656,   1959552,    279936;
  8:  1679616, 13436928,  47029248,  94058496,  117573120,   94058496,  47029248,  13436928,  1679616;
  9: 10077696, 90699264, 362797056, 846526464, 1269789696, 1269789696, 846526464, 362797056, 90699264, 10077696;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+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[(6 + 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] = (6 + 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-86b1b19edf92ec9a
  • 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-86b1b19edf92ec9a
  • 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-86b1b19edf92ec9a

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 (59)

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