Real-rooted atlas

A013613 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(7 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
171219
249168144361
33431764302417286859
4240116464423364838420736130321
5168071440604939208467207257602488322476099
611764912101045186160118540801524096010450944298598447045881
782354398825165082436814521248024893568025604812814631321635831808893871739
85764801790601284743607681626379776348509952047795650564096770048200658124842998169616983563041
94035360762259850842692469121707698764843912253952752781496328603217100863207309312270888468485159780352322687697779
10282475249484243284037355910480170769876480512309629440105389409484815055629926401474837217280948109639680361184624640619173642246131066257801
111977326743372867328683196005674401643660061120563540592384013524974217216231856700866562839061643264024334814085120139056080486404767637045248743008370688116490258898219
1213841287201284735050992268464476649615340827237120591717622003201622996906065923245993812131844769623560683525110310957875203893570253619202002407559004166241270313779289161004482562213314919066161
1396889010407215924080335622209333977376139601527857792598292262247680184615898064998442197919557713927233929067036672930076594333286488578723269836806073969595645952283977799276953681136514079129610699320537907242052983462257059
14678223072849162773537483761813762274819041243722702733056586326417002726420102620011522048516924514581995521012750069385134081519125104077701121736142976088801281488122550933258249276608109713817639756891898773504104853341271490561283918464548864799006685782884121

Generated b-files: A001029 (row sums), A038278, A038333 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-96d5ef3a0d05e14f · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-48a0cbb267bcb941

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:        7,        12;
  2:       49,       168,        144;
  3:      343,      1764,       3024,        1728;
  4:     2401,     16464,      42336,       48384,       20736;
  5:    16807,    144060,     493920,      846720,      725760,      248832;
  6:   117649,   1210104,    5186160,    11854080,    15240960,    10450944,     2985984;
  7:   823543,   9882516,   50824368,   145212480,   248935680,   256048128,   146313216,    35831808;
  8:  5764801,  79060128,  474360768,  1626379776,  3485099520,  4779565056,  4096770048,  2006581248,   429981696;
  9: 40353607, 622598508, 4269246912, 17076987648, 43912253952, 75278149632, 86032171008, 63207309312, 27088846848, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (7+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[(7 + 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] = (7 + 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.

A038278: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        7,        12;
  2:       49,       168,        144;
  3:      343,      1764,       3024,        1728;
  4:     2401,     16464,      42336,       48384,       20736;
  5:    16807,    144060,     493920,      846720,      725760,      248832;
  6:   117649,   1210104,    5186160,    11854080,    15240960,    10450944,     2985984;
  7:   823543,   9882516,   50824368,   145212480,   248935680,   256048128,   146313216,    35831808;
  8:  5764801,  79060128,  474360768,  1626379776,  3485099520,  4779565056,  4096770048,  2006581248,   429981696;
  9: 40353607, 622598508, 4269246912, 17076987648, 43912253952, 75278149632, 86032171008, 63207309312, 27088846848, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (7+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[(7 + 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] = (7 + 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.

A038333: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         12,           7;
  2:        144,         168,          49;
  3:       1728,        3024,        1764,         343;
  4:      20736,       48384,       42336,       16464,        2401;
  5:     248832,      725760,      846720,      493920,      144060,       16807;
  6:    2985984,    10450944,    15240960,    11854080,     5186160,     1210104,      117649;
  7:   35831808,   146313216,   256048128,   248935680,   145212480,    50824368,     9882516,     823543;
  8:  429981696,  2006581248,  4096770048,  4779565056,  3485099520,  1626379776,   474360768,   79060128,   5764801;
  9: 5159780352, 27088846848, 63207309312, 86032171008, 75278149632, 43912253952, 17076987648, 4269246912, 622598508, 40353607;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+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[(7 + 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] = (7 + 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-64a82ef46e481e9c
  • 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-64a82ef46e481e9c
  • 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-64a82ef46e481e9c

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 (3)

Equal up to content, sign, reversal and a small index shift.

References

  1. [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