Real-rooted atlas

A013610 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(4 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141216
21696144256
364576172817284096
4256307213824276482073665536
5102415360921602764804147202488321048576
6409673728552960221184049766405971968298598416777216
71638434406430965761548288046448640836075528360755235831808268435456

Show all 15 stored rows

Generated b-files: A001025 (row sums), A038242, A038330 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-63e0d27d213f2270 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f9762973ec1089d7

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:      4,      12;
  2:     16,      96,      144;
  3:     64,     576,     1728,      1728;
  4:    256,    3072,    13824,     27648,      20736;
  5:   1024,   15360,    92160,    276480,     414720,     248832;
  6:   4096,   73728,   552960,   2211840,    4976640,    5971968,     2985984;
  7:  16384,  344064,  3096576,  15482880,   46448640,   83607552,    83607552,    35831808;
  8:  65536, 1572864, 16515072,  99090432,  371589120,  891813888,  1337720832,  1146617856,   429981696;
  9: 262144, 7077888, 84934656, 594542592, 2675441664, 8026324992, 16052649984, 20639121408, 15479341056, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (4+12*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[(4 + 12*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] = (4 + 12*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.

A038242: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,      12;
  2:     16,      96,      144;
  3:     64,     576,     1728,      1728;
  4:    256,    3072,    13824,     27648,      20736;
  5:   1024,   15360,    92160,    276480,     414720,     248832;
  6:   4096,   73728,   552960,   2211840,    4976640,    5971968,     2985984;
  7:  16384,  344064,  3096576,  15482880,   46448640,   83607552,    83607552,    35831808;
  8:  65536, 1572864, 16515072,  99090432,  371589120,  891813888,  1337720832,  1146617856,   429981696;
  9: 262144, 7077888, 84934656, 594542592, 2675441664, 8026324992, 16052649984, 20639121408, 15479341056, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (4+12*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[(4 + 12*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] = (4 + 12*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.

A038330: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         12,           4;
  2:        144,          96,          16;
  3:       1728,        1728,         576,          64;
  4:      20736,       27648,       13824,        3072,        256;
  5:     248832,      414720,      276480,       92160,      15360,       1024;
  6:    2985984,     5971968,     4976640,     2211840,     552960,      73728,      4096;
  7:   35831808,    83607552,    83607552,    46448640,   15482880,    3096576,    344064,    16384;
  8:  429981696,  1146617856,  1337720832,   891813888,  371589120,   99090432,  16515072,  1572864,   65536;
  9: 5159780352, 15479341056, 20639121408, 16052649984, 8026324992, 2675441664, 594542592, 84934656, 7077888, 262144;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (4+12*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[(4 + 12*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] = (4 + 12*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-82b80af703a32f5e
  • 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-82b80af703a32f5e
  • 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-82b80af703a32f5e

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 (16)

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