A038271 → 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

\(P_{n}(x) = \left(12 + 5\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	12	5							17
2	144	120	25						289
3	1728	2160	900	125					4913
4	20736	34560	21600	6000	625				83521
5	248832	518400	432000	180000	37500	3125			1419857
6	2985984	7464960	7776000	4320000	1350000	225000	15625		24137569
7	35831808	104509440	130636800	90720000	37800000	9450000	1312500	78125	410338673

Show all 15 stored rows

Generated b-files: A001026 (row sums), A038254 (reversed triangle), A038331 ?

Computed in your browser ?

Technical details

Node id: n-7d034d48b0ca9aab · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-f055f948fcf55643

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 ?

A038254: reversed triangle, 33 terms agree, 7 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
A001026: 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 ?
A038331: triangle, 36 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 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:         12,           5;
  2:        144,         120,          25;
  3:       1728,        2160,         900,         125;
  4:      20736,       34560,       21600,        6000,         625;
  5:     248832,      518400,      432000,      180000,       37500,       3125;
  6:    2985984,     7464960,     7776000,     4320000,     1350000,     225000,      15625;
  7:   35831808,   104509440,   130636800,    90720000,    37800000,    9450000,    1312500,     78125;
  8:  429981696,  1433272320,  2090188800,  1741824000,   907200000,  302400000,   63000000,   7500000,   390625;
  9: 5159780352, 19349176320, 32248627200, 31352832000, 19595520000, 8164800000, 2268000000, 405000000, 42187500, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (12+5*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[(12 + 5*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] = (12 + 5*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.

A038254: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:       1;
  1:       5,       12;
  2:      25,      120,       144;
  3:     125,      900,      2160,       1728;
  4:     625,     6000,     21600,      34560,      20736;
  5:    3125,    37500,    180000,     432000,     518400,      248832;
  6:   15625,   225000,   1350000,    4320000,    7776000,     7464960,     2985984;
  7:   78125,  1312500,   9450000,   37800000,   90720000,   130636800,   104509440,    35831808;
  8:  390625,  7500000,  63000000,  302400000,  907200000,  1741824000,  2090188800,  1433272320,   429981696;
  9: 1953125, 42187500, 405000000, 2268000000, 8164800000, 19595520000, 31352832000, 32248627200, 19349176320, 5159780352;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (12+5*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[(12 + 5*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] = (12 + 5*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.

A038331: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:          1;
  1:         12,           5;
  2:        144,         120,          25;
  3:       1728,        2160,         900,         125;
  4:      20736,       34560,       21600,        6000,         625;
  5:     248832,      518400,      432000,      180000,       37500,       3125;
  6:    2985984,     7464960,     7776000,     4320000,     1350000,     225000,      15625;
  7:   35831808,   104509440,   130636800,    90720000,    37800000,    9450000,    1312500,     78125;
  8:  429981696,  1433272320,  2090188800,  1741824000,   907200000,  302400000,   63000000,   7500000,   390625;
  9: 5159780352, 19349176320, 32248627200, 31352832000, 19595520000, 8164800000, 2268000000, 405000000, 42187500, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (12+5*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[(12 + 5*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] = (12 + 5*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-7ab50ded9e353f77
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-7ab50ded9e353f77
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-7ab50ded9e353f77

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 A007318 e-7ab50ded9e353f77

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-8a6d49b40ba3d6fa root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-cb24020f7808d9ac root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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