Real-rooted atlas

A038327 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(13 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113114
2169261196
321975073912744
4285618788101452138416
5371293142805219701690651537824
6482680922277584284154394025357817529536
762748517337876637797153999635768953549911105413504
8815730721501988136135150652207924081999270123032473210411475789056
910604499373734157648922589466124054519564678291835986861845486084117120661046784
10137858491849106044993730367078824457529822040101362989093565836599781026364076051301289254654976
11179216039403715164434103395832474655151345955689652070701061022299857581715373669425130362505929514314049565169664
122329808512248121505924728444909866046203423329898620604037867068954969682546444599715162940640561413769548334011154156156693912375296
133028751065922533028751065922531397885107348863942752866881475822170516951049845437927107676455172828280424447785409120421115628342131821691793714773254144
14393737638569928942402514922915422120125746145771652346383429468137996350340849212302077447462449639355163215352910344144949074277433285862858956179970815379182111112006825558016

Generated b-files: A001023 (row sums), A123187 (reversed triangle), A133371 ?

Computed in your browser ?

Technical details
Node id
n-7a809bc95dc4396c · revision 1 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-96a9640a0a811b71

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:          13,          1;
  2:         169,         26,          1;
  3:        2197,        507,         39,         1;
  4:       28561,       8788,       1014,        52,        1;
  5:      371293,     142805,      21970,      1690,       65,       1;
  6:     4826809,    2227758,     428415,     43940,     2535,      78,      1;
  7:    62748517,   33787663,    7797153,    999635,    76895,    3549,     91,    1;
  8:   815730721,  501988136,  135150652,  20792408,  1999270,  123032,   4732,  104,   1;
  9: 10604499373, 7341576489, 2258946612, 405451956, 46782918, 3598686, 184548, 6084, 117, 1;
%F Formula
The row polynomials satisfy P(n,x) = (13+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[(13 + 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] = (13 + 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.

A123187: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  13;
  2: 1,  26,  169;
  3: 1,  39,  507,   2197;
  4: 1,  52, 1014,   8788,   28561;
  5: 1,  65, 1690,  21970,  142805,   371293;
  6: 1,  78, 2535,  43940,  428415,  2227758,   4826809;
  7: 1,  91, 3549,  76895,  999635,  7797153,  33787663,   62748517;
  8: 1, 104, 4732, 123032, 1999270, 20792408, 135150652,  501988136,  815730721;
  9: 1, 117, 6084, 184548, 3598686, 46782918, 405451956, 2258946612, 7341576489, 10604499373;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (13+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[(13 + 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] = (13 + 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.

A133371: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:           1;
  1:          13,          1;
  2:         169,         26,          1;
  3:        2197,        507,         39,         1;
  4:       28561,       8788,       1014,        52,        1;
  5:      371293,     142805,      21970,      1690,       65,       1;
  6:     4826809,    2227758,     428415,     43940,     2535,      78,      1;
  7:    62748517,   33787663,    7797153,    999635,    76895,    3549,     91,    1;
  8:   815730721,  501988136,  135150652,  20792408,  1999270,  123032,   4732,  104,   1;
  9: 10604499373, 7341576489, 2258946612, 405451956, 46782918, 3598686, 184548, 6084, 117, 1;
%F Formula
The row polynomials satisfy P(n,x) = (13+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[(13 + 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] = (13 + 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-d06691736a5246b1
  • 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-d06691736a5246b1
  • 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-d06691736a5246b1

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

No candidate equalities or related sequences.

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