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 ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	7	12							19
2	49	168	144						361
3	343	1764	3024	1728					6859
4	2401	16464	42336	48384	20736				130321
5	16807	144060	493920	846720	725760	248832			2476099
6	117649	1210104	5186160	11854080	15240960	10450944	2985984		47045881
7	823543	9882516	50824368	145212480	248935680	256048128	146313216	35831808	893871739

Show all 15 stored rows

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 ?

A038278: triangle, 32 terms agree, 7 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
A038333: reversed triangle, 32 terms agree, 7 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A001029: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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

basis_map A013609 e-64a82ef46e481e9c

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle of coefficients in expansion of (1+6x)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-1784cac1b85d5f84 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-5658c7be1cbffd3f root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

References

[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