A142459: Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).

A142459 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 A142459

\(P_{n}(x) = \left(1 - 7\,x + 4\,n x\right)\,P_{n-1}(x) + \left(4\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	1	10	1													12
4	1	59	59	1												120
5	1	308	1062	308	1											1680
6	1	1557	13562	13562	1557	1										30240
7	1	7806	148527	352612	148527	7806	1									665280
8	1	39055	1500669	7108915	7108915	1500669	39055	1								17297280
9	1	195304	14482396	123929944	241703110	123929944	14482396	195304	1							518918400
10	1	976553	136005380	1973149172	6711481694	6711481694	1973149172	136005380	976553	1						17643225600
11	1	4882802	1256274669	29595095256	163423918098	281882231148	163423918098	29595095256	1256274669	4882802	1					670442572800
12	1	24414051	11487135695	426193302405	3636464370090	10005124806558	10005124806558	3636464370090	426193302405	11487135695	24414051	1				28158588057600
13	1	122070300	104385197346	5965536951980	75884273270895	315565087670328	500256240327900	315565087670328	75884273270895	5965536951980	104385197346	122070300	1			1295295050649600
14	1	610351549	944959939614	81831773466926	1510757512828475	9131047859016423	21657793550637012	21657793550637012	9131047859016423	1510757512828475	81831773466926	944959939614	610351549	1		64764752532480000
15	1	3051757798	8534546682427	1106336252352668	29037980430228041	247650033013998458	842769418113467259	1256152025936946696	842769418113467259	247650033013998458	29037980430228041	1106336252352668	8534546682427	3051757798	1	3497296636753920000

Generated b-files: A001813 (row sums), A142459, A142459 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-d023ddeb9da11712 · revision 1 · JSON
Definition source: oeis_catalog:A142459
Fingerprint: f6-127028c66ab36cef

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 -1)
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 ?

A142459: 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 ?
A001813: 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 ?
A142459: 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,     10,         1;
   4: 1,     59,        59,          1;
   5: 1,    308,      1062,        308,          1;
   6: 1,   1557,     13562,      13562,       1557,          1;
   7: 1,   7806,    148527,     352612,     148527,       7806,          1;
   8: 1,  39055,   1500669,    7108915,    7108915,    1500669,      39055,         1;
   9: 1, 195304,  14482396,  123929944,  241703110,  123929944,   14482396,    195304,      1;
  10: 1, 976553, 136005380, 1973149172, 6711481694, 6711481694, 1973149172, 136005380, 976553, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*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 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {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 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A142459: triangle, offset 1, from the row P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,     10,         1;
   4: 1,     59,        59,          1;
   5: 1,    308,      1062,        308,          1;
   6: 1,   1557,     13562,      13562,       1557,          1;
   7: 1,   7806,    148527,     352612,     148527,       7806,          1;
   8: 1,  39055,   1500669,    7108915,    7108915,    1500669,      39055,         1;
   9: 1, 195304,  14482396,  123929944,  241703110,  123929944,   14482396,    195304,      1;
  10: 1, 976553, 136005380, 1973149172, 6711481694, 6711481694, 1973149172, 136005380, 976553, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*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 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*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 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*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.

A142459: reversed triangle, offset 1, from the row P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,     10,         1;
   4: 1,     59,        59,          1;
   5: 1,    308,      1062,        308,          1;
   6: 1,   1557,     13562,      13562,       1557,          1;
   7: 1,   7806,    148527,     352612,     148527,       7806,          1;
   8: 1,  39055,   1500669,    7108915,    7108915,    1500669,      39055,         1;
   9: 1, 195304,  14482396,  123929944,  241703110,  123929944,   14482396,    195304,      1;
  10: 1, 976553, 136005380, 1973149172, 6711481694, 6711481694, 1973149172, 136005380, 976553, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*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 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*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 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*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 >= 1, 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-ce13bfb75f31b51b
Coefficients of one sign. Every nonzero row, n >= 1: 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-ce13bfb75f31b51b
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-ce13bfb75f31b51b
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-ce13bfb75f31b51b

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-ce13bfb75f31b51b root: theorem check

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

Used in 3 proof steps

e-e113043a8d357c9d gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n…
e-2762452c13f28e99 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3…
e-2762452c13f28e99 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3…

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A257618: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.

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