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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	429981696	1433272320	2090188800	1741824000	907200000	302400000	63000000	7500000	390625							6975757441
9	5159780352	19349176320	32248627200	31352832000	19595520000	8164800000	2268000000	405000000	42187500	1953125						118587876497
10	61917364224	257989017600	483729408000	537477120000	391910400000	195955200000	68040000000	16200000000	2531250000	234375000	9765625					2015993900449
11	743008370688	3405455032320	7094697984000	8868372480000	7390310400000	4311014400000	1796256000000	534600000000	111375000000	15468750000	1289062500	48828125				34271896307633
12	8916100448256	44580502241280	102163650969600	141893959680000	133025587200000	88683724800000	43110144000000	15396480000000	4009500000000	742500000000	92812500000	7031250000	244140625			582622237229761
13	106993205379072	579546529136640	1448866322841600	2213545771008000	2305776844800000	1729332633600000	960740352000000	400308480000000	125096400000000	28957500000000	4826250000000	548437500000	38085937500	1220703125		9904578032905937
14	1283918464548864	7489524376535040	20284128519782400	33806880866304000	38737050992640000	32280875827200000	20175547392000000	9607403520000000	3502699200000000	972972000000000	202702500000000	30712500000000	3199218750000	205078125000	6103515625	168377826559400929

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