A142461: Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = 6.

A142461 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A142461

\(P_{n}(x) = \left(1 - 5\,x + 6\,n x\right)\,P_{n-1}(x) + \left(6\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	1	14	1													16
3	1	111	111	1												224
4	1	796	2886	796	1											4480
5	1	5597	52642	52642	5597	1										116480
6	1	39210	824271	2000396	824271	39210	1									3727360
7	1	274507	11931033	58614299	58614299	11931033	274507	1								141639680
8	1	1921592	165260188	1483533704	2930714950	1483533704	165260188	1921592	1							6232145920
9	1	13451193	2231010900	34301767332	119257418574	119257418574	34301767332	2231010900	13451193	1						311607296000
10	1	94158406	29662250157	747667048008	4250600855634	7393959951588	4250600855634	747667048008	29662250157	94158406	1					17450008576000
11	1	659108903	390787964371	15659124169845	138414704455194	386484990157686	386484990157686	138414704455194	15659124169845	390787964371	659108903	1				1081900531712000
12	1	4613762388	5120449179906	319016697267460	4227664695702255	17932866986461608	28599889271668764	17932866986461608	4227664695702255	319016697267460	5120449179906	4613762388	1			73569236156416000
13	1	32296336789	66874961418774	6373664648056006	123237535742266675	763074446669720343	1829309183469593412	1829309183469593412	763074446669720343	123237535742266675	6373664648056006	66874961418774	32296336789	1		5444123475574784000
14	1	226074357602	871732131029659	125580250728121972	3469731937088083241	30433372312585997758	105075087675191253051	157320589778385033432	105075087675191253051	30433372312585997758	3469731937088083241	125580250728121972	871732131029659	226074357602	1	435529878045982720000

Generated b-files: A047657 (row sums), A142461, A142461 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-3bc26d3108ee6bfd · revision 1 · JSON
Definition source: oeis_catalog:A142461
Fingerprint: f6-b62324396409c257

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A142461: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A142461: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A047657: 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: 1,        1;
  2: 1,       14,          1;
  3: 1,      111,        111,           1;
  4: 1,      796,       2886,         796,            1;
  5: 1,     5597,      52642,       52642,         5597,            1;
  6: 1,    39210,     824271,     2000396,       824271,        39210,           1;
  7: 1,   274507,   11931033,    58614299,     58614299,     11931033,      274507,          1;
  8: 1,  1921592,  165260188,  1483533704,   2930714950,   1483533704,   165260188,    1921592,        1;
  9: 1, 13451193, 2231010900, 34301767332, 119257418574, 119257418574, 34301767332, 2231010900, 13451193, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(6*n-5)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx 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[(1 + (6*n - 5)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], 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] = (1 + (6*n - 5)*x)*v[i - 1] + (6*x - 6*x^2)*deriv(v[i - 1], x)); 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.

A142461: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       14,          1;
   4: 1,      111,        111,           1;
   5: 1,      796,       2886,         796,            1;
   6: 1,     5597,      52642,       52642,         5597,            1;
   7: 1,    39210,     824271,     2000396,       824271,        39210,           1;
   8: 1,   274507,   11931033,    58614299,     58614299,     11931033,      274507,          1;
   9: 1,  1921592,  165260188,  1483533704,   2930714950,   1483533704,   165260188,    1921592,        1;
  10: 1, 13451193, 2231010900, 34301767332, 119257418574, 119257418574, 34301767332, 2231010900, 13451193, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(6*n-11)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (6*n - 11)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (6*n - 11)*x)*v[i - 1] + (6*x - 6*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A142461: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       14,          1;
   4: 1,      111,        111,           1;
   5: 1,      796,       2886,         796,            1;
   6: 1,     5597,      52642,       52642,         5597,            1;
   7: 1,    39210,     824271,     2000396,       824271,        39210,           1;
   8: 1,   274507,   11931033,    58614299,     58614299,     11931033,      274507,          1;
   9: 1,  1921592,  165260188,  1483533704,   2930714950,   1483533704,   165260188,    1921592,        1;
  10: 1, 13451193, 2231010900, 34301767332, 119257418574, 119257418574, 34301767332, 2231010900, 13451193, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(6*n-11)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (6*n - 11)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (6*n - 11)*x)*v[i - 1] + (6*x - 6*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-f43e8fb782fe1ff0
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-f43e8fb782fe1ff0
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-f43e8fb782fe1ff0
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-f43e8fb782fe1ff0

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

theorem_check liu_wang e-f43e8fb782fe1ff0 root: theorem check

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 3 proof steps

e-de18761c144e2c93 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n)…
e-5de672ddb1a17fcf obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) =…
e-5de672ddb1a17fcf obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) =…

Related sequences ?

No candidate equalities or related sequences.

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