A167884: Triangle read by rows: T(n,k) 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 = 8.

A167884 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 A167884

\(P_{n}(x) = \left(1 - 7\,x + 8\,n x\right)\,P_{n-1}(x) + \left(8\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)

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

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	18	1													20
3	1	179	179	1												360
4	1	1636	6086	1636	1											9360
5	1	14757	144362	144362	14757	1										318240
6	1	132854	2941135	7218100	2941135	132854	1									13366080
7	1	1195735	55446309	277509955	277509955	55446309	1195735	1								668304000
8	1	10761672	1001178268	9211047544	18315657030	9211047544	1001178268	10761672	1							38761632000
9	1	96855113	17633445860	279333923732	982069631294	982069631294	279333923732	17633445860	96855113	1						2558267712000
10	1	871696090	306064161965	7988454507320	46095660095570	80529709766108	46095660095570	7988454507320	306064161965	871696090	1					189311810688000
11	1	7845264891	5266724567975	219605533210725	1976498690071050	5560405445093358	5560405445093358	1976498690071050	219605533210725	5266724567975	7845264891	1				15523568476416000
12	1	70607384108	90169784111746	5874609223730300	79498816431041775	340637048582877528	544919733619149084	340637048582877528	79498816431041775	5874609223730300	90169784111746	70607384108	1			1397121162877440000
13	1	635466457069	1539170387085294	154168983106308926	3052307415556690475	19133542059915694023	46117378716562324212	46117378716562324212	19133542059915694023	3052307415556690475	154168983106308926	1539170387085294	635466457069	1		136917873961989120000
14	1	5719198113726	26227536826785691	3991210742108314316	113213832344981808681	1007293665792181859618	3503431791006073997883	5257381173688104960168	3503431791006073997883	1007293665792181859618	113213832344981808681	3991210742108314316	26227536826785691	5719198113726	1	14513294639970846720000

Generated b-files: A084948 (row sums), A167884, A167884 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-90af30f849fc554d · revision 1 · JSON
Definition source: oeis_catalog:A167884
Fingerprint: f6-096b716e28fb0b30

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 ?

A167884: 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 ?
A084948: 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 ?
A167884: 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 ?

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,       18,           1;
  3: 1,      179,         179,            1;
  4: 1,     1636,        6086,         1636,            1;
  5: 1,    14757,      144362,       144362,        14757,            1;
  6: 1,   132854,     2941135,      7218100,      2941135,       132854,            1;
  7: 1,  1195735,    55446309,    277509955,    277509955,     55446309,      1195735,           1;
  8: 1, 10761672,  1001178268,   9211047544,  18315657030,   9211047544,   1001178268,    10761672,        1;
  9: 1, 96855113, 17633445860, 279333923732, 982069631294, 982069631294, 279333923732, 17633445860, 96855113, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(8*n-7)*x)*P(n-1,x) + (8*x-8*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 + (8*n - 7)*x)*P[n - 1, x] + (8*x - 8*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 + (8*n - 7)*x)*v[i - 1] + (8*x - 8*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.

A167884: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       18,           1;
   4: 1,      179,         179,            1;
   5: 1,     1636,        6086,         1636,            1;
   6: 1,    14757,      144362,       144362,        14757,            1;
   7: 1,   132854,     2941135,      7218100,      2941135,       132854,            1;
   8: 1,  1195735,    55446309,    277509955,    277509955,     55446309,      1195735,           1;
   9: 1, 10761672,  1001178268,   9211047544,  18315657030,   9211047544,   1001178268,    10761672,        1;
  10: 1, 96855113, 17633445860, 279333923732, 982069631294, 982069631294, 279333923732, 17633445860, 96855113, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(8*n-15)*x)*P(n-1,x) + (8*x-8*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 + (8*n - 15)*x)*P[n - 1, x] + (8*x - 8*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 + (8*n - 15)*x)*v[i - 1] + (8*x - 8*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.

A167884: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       18,           1;
   4: 1,      179,         179,            1;
   5: 1,     1636,        6086,         1636,            1;
   6: 1,    14757,      144362,       144362,        14757,            1;
   7: 1,   132854,     2941135,      7218100,      2941135,       132854,            1;
   8: 1,  1195735,    55446309,    277509955,    277509955,     55446309,      1195735,           1;
   9: 1, 10761672,  1001178268,   9211047544,  18315657030,   9211047544,   1001178268,    10761672,        1;
  10: 1, 96855113, 17633445860, 279333923732, 982069631294, 982069631294, 279333923732, 17633445860, 96855113, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(8*n-15)*x)*P(n-1,x) + (8*x-8*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 + (8*n - 15)*x)*P[n - 1, x] + (8*x - 8*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 + (8*n - 15)*x)*v[i - 1] + (8*x - 8*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-46d699555fa04487
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-46d699555fa04487
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-46d699555fa04487
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-46d699555fa04487

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-46d699555fa04487 root: theorem check

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

Used in 3 proof steps

e-15b23c3dbd64ff2a gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwis…
e-c70944d6f4a4968f obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, o…
e-c70944d6f4a4968f obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, o…

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