Real-rooted atlas

A013620: Triangle of coefficients in expansion of (2+3x)^n.

A013620 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · all roots real, in [−1, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A013620
\(P_{n}(x) = \left(2 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
2412925
38365427125
4169621621681625
53224072010808102433125
664576216043204860291672915625
71281344604815120226802041210206218778125
8256307216128483849072010886481648349926561390625
9512691241472145152326592489888489888314928118098196831953125
1010241536010368041472010886401959552244944020995201180980393660590499765625
11204833792253440114048034214407185024107775361154736086605204330260129907817714748828125
124096737286082563041280102643202463436843110144554273285196312034642080155889364251528531441244140625
138192159744143769679073282965248080061696160123392240185088270208224225173520135104112552698641381746615943231220703125
141638434406433546242012774483026944249080832560431872960740352126097171212609717129457287845158520641934445244464104447829696103515625

Generated b-files: A000351 (row sums), A013620, A038220 (reversed triangle) ?

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 5\,x + 3\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-5733d79f45a14084 · revision 2 · JSON
Definition source
oeis_catalog:A013620
derived:mobius_x_over_x_plus_one (equal by e-49cf64315b8edeba)
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:   2,    3;
  2:   4,   12,     9;
  3:   8,   36,    54,     27;
  4:  16,   96,   216,    216,     81;
  5:  32,  240,   720,   1080,    810,    243;
  6:  64,  576,  2160,   4320,   4860,   2916,    729;
  7: 128, 1344,  6048,  15120,  22680,  20412,  10206,   2187;
  8: 256, 3072, 16128,  48384,  90720, 108864,  81648,  34992,   6561;
  9: 512, 6912, 41472, 145152, 326592, 489888, 489888, 314928, 118098, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (2+3*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[(2 + 3*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] = (2 + 3*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.

A013620: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,    3;
  2:   4,   12,     9;
  3:   8,   36,    54,     27;
  4:  16,   96,   216,    216,     81;
  5:  32,  240,   720,   1080,    810,    243;
  6:  64,  576,  2160,   4320,   4860,   2916,    729;
  7: 128, 1344,  6048,  15120,  22680,  20412,  10206,   2187;
  8: 256, 3072, 16128,  48384,  90720, 108864,  81648,  34992,   6561;
  9: 512, 6912, 41472, 145152, 326592, 489888, 489888, 314928, 118098, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (2+3*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[(2 + 3*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] = (2 + 3*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.

A038220: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,      2;
  2:     9,     12,      4;
  3:    27,     54,     36,      8;
  4:    81,    216,    216,     96,     16;
  5:   243,    810,   1080,    720,    240,     32;
  6:   729,   2916,   4860,   4320,   2160,    576,     64;
  7:  2187,  10206,  20412,  22680,  15120,   6048,   1344,   128;
  8:  6561,  34992,  81648, 108864,  90720,  48384,  16128,  3072,  256;
  9: 19683, 118098, 314928, 489888, 489888, 326592, 145152, 41472, 6912, 512;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+3*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[(2 + 3*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] = (2 + 3*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-0ecfea69e82c3845
  • 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-0ecfea69e82c3845, e-31e5a5a6b5727160, e-57079e0e5fed1718, e-cb393d3715764915, e-e9dcc990956c90e9
  • 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-0ecfea69e82c3845, e-31e5a5a6b5727160, e-57079e0e5fed1718, e-cb393d3715764915, e-e9dcc990956c90e9
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-0ecfea69e82c3845
  • 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-31e5a5a6b5727160, e-cb393d3715764915, e-e9dcc990956c90e9
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-6f75aea89a6e01af
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 2] to roots in (-inf, 0]: an input root r <= 2 gives the root (r - a)/d <= 0; a larger root gives a positive root
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-6f75aea89a6e01af
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-57079e0e5fed1718

Proof chains

7 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

Used in 18 proof steps

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