A144440: Triangle T(n,k) by rows: T(n, k) = (3n-3k+1)T(n-1, k-1) +(3k-2)T(n-1, k) + 3T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

A144440 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 A144440

\(P_{n}(x) = \left(1 - 2\,x + 3\,n x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2}\right)\,P_{n-1}'(x) + \left(3\,x\right)\,P_{n-2}(x)\)

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

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

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	11	1													13
3	1	54	54	1												110
4	1	229	789	229	1											1249
5	1	932	7975	7975	932	1										17816
6	1	3747	68628	161867	68628	3747	1									306619
7	1	15010	543144	2534759	2534759	543144	15010	1								6185828
8	1	60065	4098439	34243778	66389335	34243778	4098439	60065	1							143193901
9	1	240288	30055533	421937553	1418566080	1418566080	421937553	30055533	240288	1						3741598910
10	1	961183	216576126	4892892573	26560903881	45593282565	26560903881	4892892573	216576126	961183	1					108935950093
11	1	3844766	1543666870	54433495479	454201199718	1238405393019	1238405393019	454201199718	54433495479	1543666870	3844766	1				3497175199706
12	1	15379101	10927739385	588207355528	7280131661028	29886595393743	47196184782417	29886595393743	7280131661028	588207355528	10927739385	15379101	1			122727939839989
13	1	61516444	77028599427	6225464476825	111274818034585	661551421424742	1557947825707326	1557947825707326	661551421424742	111274818034585	6225464476825	77028599427	61516444	1		4674153239518700
14	1	246065819	541522441720	64906400366923	1641326655297764	13722358042747336	46229454010238973	68691292885469595	46229454010238973	13722358042747336	1641326655297764	64906400366923	541522441720	246065819	1	192008466639786667

Generated b-files: A144440, A144440 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-ef0a5df5e949a196 · revision 1 · JSON
Definition source: oeis_catalog:A144440
Fingerprint: f6-28fef730de5cdff7

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 ?

A144440: 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 ?
A144440: 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,     11,        1;
  3: 1,     54,       54,         1;
  4: 1,    229,      789,       229,          1;
  5: 1,    932,     7975,      7975,        932,          1;
  6: 1,   3747,    68628,    161867,      68628,       3747,         1;
  7: 1,  15010,   543144,   2534759,    2534759,     543144,     15010,        1;
  8: 1,  60065,  4098439,  34243778,   66389335,   34243778,   4098439,    60065,      1;
  9: 1, 240288, 30055533, 421937553, 1418566080, 1418566080, 421937553, 30055533, 240288, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(3*n-2)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) + 3*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 + (3*n - 2)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x] + 3*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 + (3*n - 2)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x) + 3*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.

A144440: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,     11,        1;
   4: 1,     54,       54,         1;
   5: 1,    229,      789,       229,          1;
   6: 1,    932,     7975,      7975,        932,          1;
   7: 1,   3747,    68628,    161867,      68628,       3747,         1;
   8: 1,  15010,   543144,   2534759,    2534759,     543144,     15010,        1;
   9: 1,  60065,  4098439,  34243778,   66389335,   34243778,   4098439,    60065,      1;
  10: 1, 240288, 30055533, 421937553, 1418566080, 1418566080, 421937553, 30055533, 240288, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(3*n-5)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) + 3*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 + (3*n - 5)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x] + 3*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 + (3*n - 5)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x) + 3*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.

A144440: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,     11,        1;
   4: 1,     54,       54,         1;
   5: 1,    229,      789,       229,          1;
   6: 1,    932,     7975,      7975,        932,          1;
   7: 1,   3747,    68628,    161867,      68628,       3747,         1;
   8: 1,  15010,   543144,   2534759,    2534759,     543144,     15010,        1;
   9: 1,  60065,  4098439,  34243778,   66389335,   34243778,   4098439,    60065,      1;
  10: 1, 240288, 30055533, 421937553, 1418566080, 1418566080, 421937553, 30055533, 240288, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(3*n-5)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) + 3*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 + (3*n - 5)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x] + 3*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 + (3*n - 5)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x) + 3*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-8f7192735b9d8398
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-8f7192735b9d8398
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-8f7192735b9d8398
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-8f7192735b9d8398

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-8f7192735b9d8398 root: theorem check

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

Used in 1 proof step

e-53cca568857b5184 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T…

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