A144439: Triangle T(n, k) = (m(n-k) + 1)T(n-1, k-1) + (m(k-1) + 1)T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.

A144439 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A144439

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

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

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

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	8	1													10
3	1	31	31	1												64
4	1	102	326	102	1											532
5	1	317	2406	2406	317	1										5448
6	1	964	15087	34336	15087	964	1									66440
7	1	2907	86673	380947	380947	86673	2907	1								941056
8	1	8738	473084	3650206	6925718	3650206	473084	8738	1							15189776
9	1	26233	2502304	31874880	103245622	103245622	31874880	2502304	26233	1						275298080
10	1	78720	12974957	261604888	1350884450	2285255120	1350884450	261604888	12974957	78720	1					5536341152
11	1	236183	66422931	2056813093	16145783130	42905795414	42905795414	16145783130	2056813093	66422931	236183	1				122350101504
12	1	708574	337231938	15685677254	180801080527	716852265404	1120121191004	716852265404	180801080527	15685677254	337231938	708574	1			2947475118400
13	1	2125749	1702929258	117014457338	1929351218755	10991284854663	25400171054940	25400171054940	10991284854663	1929351218755	117014457338	1702929258	2125749	1		76879053281408
14	1	6377276	8569207163	858943038176	19852835927401	157923408718692	518487770774299	764245374030208	518487770774299	157923408718692	19852835927401	858943038176	8569207163	6377276	1	2158508442116224

Generated b-files: A144439, A144439 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-14be0ce6edbde2a1 · revision 1 · JSON
Definition source: oeis_catalog:A144439
Fingerprint: f6-50007f665083f63a

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 ?

A144439: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A144439: triangle, 55 terms agree, 10 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,     8,       1;
  3: 1,    31,      31,        1;
  4: 1,   102,     326,      102,         1;
  5: 1,   317,    2406,     2406,       317,         1;
  6: 1,   964,   15087,    34336,     15087,       964,        1;
  7: 1,  2907,   86673,   380947,    380947,     86673,     2907,       1;
  8: 1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;
  9: 1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-1)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 1)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x] + 2*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n - 1)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x) + 2*x*v[i - 2]); 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.

A144439: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,     1;
   3: 1,     8,       1;
   4: 1,    31,      31,        1;
   5: 1,   102,     326,      102,         1;
   6: 1,   317,    2406,     2406,       317,         1;
   7: 1,   964,   15087,    34336,     15087,       964,        1;
   8: 1,  2907,   86673,   380947,    380947,     86673,     2907,       1;
   9: 1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;
  10: 1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-3)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) + 2*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 3)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x] + 2*x*P[n - 2, 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; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (2*n - 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x) + 2*x*v[i - 2]); 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.

A144439: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,     1;
   3: 1,     8,       1;
   4: 1,    31,      31,        1;
   5: 1,   102,     326,      102,         1;
   6: 1,   317,    2406,     2406,       317,         1;
   7: 1,   964,   15087,    34336,     15087,       964,        1;
   8: 1,  2907,   86673,   380947,    380947,     86673,     2907,       1;
   9: 1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;
  10: 1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(2*n-3)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) + 2*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 3)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x] + 2*x*P[n - 2, 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; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (2*n - 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x) + 2*x*v[i - 2]); 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-b81aae17045623cf
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-b81aae17045623cf
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-b81aae17045623cf
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-b81aae17045623cf

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_derivative_lag e-b81aae17045623cf root: theorem check

Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Used in 1 proof step

e-9b31724594a81849 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k…

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