Real-rooted atlas

A038243 → 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
\(P_{n}(x) = \left(6 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
2364816100
3216432288641000
4129634563456153625610000
5777625920345602304076801024100000
6466561866243110402764801382403686440961000000
727993613063682612736290304019353607741441720321638410000000
81679616895795220901888278691842322432012386304412876878643265536100000000
91007769660466176161243136250822656250822656167215104743178242123366435389442621441000000000
10604661764031078401209323520214990848025082265602006581248111476736042467328010616832015728640104857610000000000
11362797056266051174488683724801773674496023648993280220723937281471492915270071091202335703040519045120692060164194304100000000000
122176782336174142586886385228185614189395968021284093952022703033548817657914982410090237132842042654720124570828802491416576301989888167772161000000000000
131306069401611319268147245277072588811067728855041844621475840221354577100819675962408961311730827264655865413632242913116160647768309761177760563213086228486710886410000000000000
14783641640967313988648963169395081216845172021657615494820397056206597605294082065976052940815740769927168918211579084840809403514881360313450496329772957696549621596165637144576268435456100000000000000

Generated b-files: A011557 (row sums), A038236 (reversed triangle), A038258 ?

Computed in your browser ?

Technical details
Node id
n-408f7506d6594551 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f06e51c139cfae05

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:        6,        4;
  2:       36,       48,        16;
  3:      216,      432,       288,        64;
  4:     1296,     3456,      3456,      1536,       256;
  5:     7776,    25920,     34560,     23040,      7680,      1024;
  6:    46656,   186624,    311040,    276480,    138240,     36864,     4096;
  7:   279936,  1306368,   2612736,   2903040,   1935360,    774144,   172032,    16384;
  8:  1679616,  8957952,  20901888,  27869184,  23224320,  12386304,  4128768,   786432,   65536;
  9: 10077696, 60466176, 161243136, 250822656, 250822656, 167215104, 74317824, 21233664, 3538944, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (6+4*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 + 4*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 + 4*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.

A038236: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       6;
  2:     16,      48,       36;
  3:     64,     288,      432,      216;
  4:    256,    1536,     3456,     3456,      1296;
  5:   1024,    7680,    23040,    34560,     25920,      7776;
  6:   4096,   36864,   138240,   276480,    311040,    186624,     46656;
  7:  16384,  172032,   774144,  1935360,   2903040,   2612736,   1306368,    279936;
  8:  65536,  786432,  4128768, 12386304,  23224320,  27869184,  20901888,   8957952,  1679616;
  9: 262144, 3538944, 21233664, 74317824, 167215104, 250822656, 250822656, 161243136, 60466176, 10077696;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+4*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 + 4*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 + 4*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.

A038258: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,        4;
  2:       36,       48,        16;
  3:      216,      432,       288,        64;
  4:     1296,     3456,      3456,      1536,       256;
  5:     7776,    25920,     34560,     23040,      7680,      1024;
  6:    46656,   186624,    311040,    276480,    138240,     36864,     4096;
  7:   279936,  1306368,   2612736,   2903040,   1935360,    774144,   172032,    16384;
  8:  1679616,  8957952,  20901888,  27869184,  23224320,  12386304,  4128768,   786432,   65536;
  9: 10077696, 60466176, 161243136, 250822656, 250822656, 167215104, 74317824, 21233664, 3538944, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (6+4*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 + 4*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 + 4*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-0e13ca82ccc1a33c
  • 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-0e13ca82ccc1a33c
  • 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-0e13ca82ccc1a33c

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 (23)

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