A038271 → 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

\(P_{n}(x) = \left(12 + 10\,x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	12	10							22
2	144	240	100						484
3	1728	4320	3600	1000					10648
4	20736	69120	86400	48000	10000				234256
5	248832	1036800	1728000	1440000	600000	100000			5153632
6	2985984	14929920	31104000	34560000	21600000	7200000	1000000		113379904
7	35831808	209018880	522547200	725760000	604800000	302400000	84000000	10000000	2494357888

Show all 15 stored rows

Generated b-files: A009966 (row sums), A038314 (reversed triangle), A038336 ?

Computed in your browser ?

Technical details

Node id: n-4f44de8e265280a7 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-da2debf48cb6165b

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 ?

A009966: 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 ?
A038314: reversed triangle, 30 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 ?
A038336: triangle, 30 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 ?

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,          10;
  2:        144,         240,          100;
  3:       1728,        4320,         3600,         1000;
  4:      20736,       69120,        86400,        48000,        10000;
  5:     248832,     1036800,      1728000,      1440000,       600000,       100000;
  6:    2985984,    14929920,     31104000,     34560000,     21600000,      7200000,      1000000;
  7:   35831808,   209018880,    522547200,    725760000,    604800000,    302400000,     84000000,    10000000;
  8:  429981696,  2866544640,   8360755200,  13934592000,  14515200000,   9676800000,   4032000000,   960000000,   100000000;
  9: 5159780352, 38698352640, 128994508800, 250822656000, 313528320000, 261273600000, 145152000000, 51840000000, 10800000000, 1000000000;

%F Formula

The row polynomials satisfy P(n,x) = (12+10*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 + 10*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 + 10*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.

A038314: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:          1;
  1:         10,          12;
  2:        100,         240,         144;
  3:       1000,        3600,        4320,         1728;
  4:      10000,       48000,       86400,        69120,        20736;
  5:     100000,      600000,     1440000,      1728000,      1036800,       248832;
  6:    1000000,     7200000,    21600000,     34560000,     31104000,     14929920,      2985984;
  7:   10000000,    84000000,   302400000,    604800000,    725760000,    522547200,    209018880,     35831808;
  8:  100000000,   960000000,  4032000000,   9676800000,  14515200000,  13934592000,   8360755200,   2866544640,   429981696;
  9: 1000000000, 10800000000, 51840000000, 145152000000, 261273600000, 313528320000, 250822656000, 128994508800, 38698352640, 5159780352;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (12+10*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 + 10*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 + 10*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.

A038336: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:          1;
  1:         12,          10;
  2:        144,         240,          100;
  3:       1728,        4320,         3600,         1000;
  4:      20736,       69120,        86400,        48000,        10000;
  5:     248832,     1036800,      1728000,      1440000,       600000,       100000;
  6:    2985984,    14929920,     31104000,     34560000,     21600000,      7200000,      1000000;
  7:   35831808,   209018880,    522547200,    725760000,    604800000,    302400000,     84000000,    10000000;
  8:  429981696,  2866544640,   8360755200,  13934592000,  14515200000,   9676800000,   4032000000,   960000000,   100000000;
  9: 5159780352, 38698352640, 128994508800, 250822656000, 313528320000, 261273600000, 145152000000, 51840000000, 10800000000, 1000000000;

%F Formula

The row polynomials satisfy P(n,x) = (12+10*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 + 10*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 + 10*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-e169cb33ba7b2c8c
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-e169cb33ba7b2c8c
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-e169cb33ba7b2c8c

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-e169cb33ba7b2c8c

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 (6)

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