Real-rooted atlas

A038221 → 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
\(P_{n}(x) = \left(6 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16915
23610881225
321697214587293375
4129677761749617496656150625
577765832017496026244019683059049759375
646656419904157464031492803542940212576453144111390625
7279936293932813226976330674404960116044641044223205224782969170859375
8167961620155392105815808317447424595213920714256704535692528229582512430467212562890625
91007769613604889681629337628570268166428310336964246550496424655046198727824232452293438742048938443359375
10604661769069926406122200320244888012806428310336011570958604814463698256012397455648069735688020232452293403486784401576650390625
113627970565986151424448961356802020326105606060978316801272805446528190920816979220455801819201534185136440767092568220230127770466313810596098649755859375
1221767823363918208204832325217689616162608844805454880485120130917131642882291049803750429456354619648276153324559201841022163728082845997367762259436291848282429536481129746337890625
131306069401625468353331222921517998081260683489894447275630871040127644203351808255288406703616382932610055424430799186312352358999321926960215399593156176881180153820722202950384551825418658283291946195068359375
14783641640961645647446016160450625986569627037559193639711529931673611913458979502082680528270387968459519132066508860311886083729286031188608372928452339145627969624673044306980169252391615117562135167295796362287679245496129192926025390625

Generated b-files: A001024 (row sums), A038263, A038296 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-9dc2047d1a16a51a · revision 1 · JSON
Definition source
derived:basis_map_A013610
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,         9;
  2:       36,       108,        81;
  3:      216,       972,      1458,        729;
  4:     1296,      7776,     17496,      17496,       6561;
  5:     7776,     58320,    174960,     262440,     196830,      59049;
  6:    46656,    419904,   1574640,    3149280,    3542940,    2125764,     531441;
  7:   279936,   2939328,  13226976,   33067440,   49601160,   44641044,   22320522,    4782969;
  8:  1679616,  20155392, 105815808,  317447424,  595213920,  714256704,  535692528,  229582512,   43046721;
  9: 10077696, 136048896, 816293376, 2857026816, 6428310336, 9642465504, 9642465504, 6198727824, 2324522934, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (6+9*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 + 9*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 + 9*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.

A038263: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,         9;
  2:       36,       108,        81;
  3:      216,       972,      1458,        729;
  4:     1296,      7776,     17496,      17496,       6561;
  5:     7776,     58320,    174960,     262440,     196830,      59049;
  6:    46656,    419904,   1574640,    3149280,    3542940,    2125764,     531441;
  7:   279936,   2939328,  13226976,   33067440,   49601160,   44641044,   22320522,    4782969;
  8:  1679616,  20155392, 105815808,  317447424,  595213920,  714256704,  535692528,  229582512,   43046721;
  9: 10077696, 136048896, 816293376, 2857026816, 6428310336, 9642465504, 9642465504, 6198727824, 2324522934, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (6+9*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 + 9*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 + 9*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.

A038296: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         9,          6;
  2:        81,        108,         36;
  3:       729,       1458,        972,        216;
  4:      6561,      17496,      17496,       7776,       1296;
  5:     59049,     196830,     262440,     174960,      58320,       7776;
  6:    531441,    2125764,    3542940,    3149280,    1574640,     419904,      46656;
  7:   4782969,   22320522,   44641044,   49601160,   33067440,   13226976,    2939328,    279936;
  8:  43046721,  229582512,  535692528,  714256704,  595213920,  317447424,  105815808,  20155392,   1679616;
  9: 387420489, 2324522934, 6198727824, 9642465504, 9642465504, 6428310336, 2857026816, 816293376, 136048896, 10077696;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+9*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 + 9*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 + 9*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-b88e4ed4246b1dea
  • 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-b88e4ed4246b1dea
  • 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-b88e4ed4246b1dea

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