A144445: Triangle, read by rows, T(n,k) = (7n-7k+1)T(n-1, k-2) + (7k-6)T(n-1, k) + 7T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

A144445 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 A144445

\(P_{n}(x) = \left(1 - 6\,x + 7\,n x\right)\,P_{n-1}(x) + \left(7\,x - 7\,x^{2}\right)\,P_{n-1}'(x) + \left(7\,x\right)\,P_{n-2}(x)\)

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

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

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	23	1													25
3	1	206	206	1												414
4	1	1677	6341	1677	1											9697
5	1	13452	133451	133451	13452	1										293808
6	1	107659	2403612	5916231	2403612	107659	1									10938775
7	1	861322	40024068	200795987	200795987	40024068	861322	1								483362756
8	1	6890633	638151479	5875203446	11687580863	5875203446	638151479	6890633	1							24728071981
9	1	55125128	9922833089	156975157885	551852740992	551852740992	156975157885	9922833089	55125128	1						1437611714190
10	1	441001095	152032863062	3954062188273	22794787701945	39815210417465	22794787701945	3954062188273	152032863062	441001095	1					93617857926217
11	1	3528008838	2309102891906	95724701168163	859850778875250	2417386415399319	2417386415399319	859850778875250	95724701168163	2309102891906	3528008838	1				6750549052686954
12	1	28224070789	34890119013753	2254790240823004	30419658989285452	130178013412051599	208173938197263689	130178013412051599	30419658989285452	2254790240823004	34890119013753	28224070789	1			533948700167752885
13	1	225792566404	525577958789703	52098747468325893	1027146759010127505	6426348000675255078	15477301717992713810	15477301717992713810	6426348000675255078	1027146759010127505	52098747468325893	525577958789703	225792566404	1		45966842057795556788
14	1	1806340531331	7903059318485408	1187411755921862751	33502050613230597076	297298858213882341292	1032737056006060594469	1549187389366652226823	1032737056006060594469	297298858213882341292	33502050613230597076	1187411755921862751	7903059318485408	1806340531331	1	4278653952276161051479

Generated b-files: A144445, A144445 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-2d05bfc763190112 · revision 1 · JSON
Definition source: oeis_catalog:A144445
Fingerprint: f6-72195b7678c08af8

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 ?

A144445: 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 ?
A144445: 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,       23,          1;
  3: 1,      206,        206,            1;
  4: 1,     1677,       6341,         1677,            1;
  5: 1,    13452,     133451,       133451,        13452,            1;
  6: 1,   107659,    2403612,      5916231,      2403612,       107659,            1;
  7: 1,   861322,   40024068,    200795987,    200795987,     40024068,       861322,          1;
  8: 1,  6890633,  638151479,   5875203446,  11687580863,   5875203446,    638151479,    6890633,        1;
  9: 1, 55125128, 9922833089, 156975157885, 551852740992, 551852740992, 156975157885, 9922833089, 55125128, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(7*n-6)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) + 7*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 + (7*n - 6)*x)*P[n - 1, x] + (7*x - 7*x^2)*D[P[n - 1, x], x] + 7*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 + (7*n - 6)*x)*v[i - 1] + (7*x - 7*x^2)*deriv(v[i - 1], x) + 7*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.

A144445: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       23,          1;
   4: 1,      206,        206,            1;
   5: 1,     1677,       6341,         1677,            1;
   6: 1,    13452,     133451,       133451,        13452,            1;
   7: 1,   107659,    2403612,      5916231,      2403612,       107659,            1;
   8: 1,   861322,   40024068,    200795987,    200795987,     40024068,       861322,          1;
   9: 1,  6890633,  638151479,   5875203446,  11687580863,   5875203446,    638151479,    6890633,        1;
  10: 1, 55125128, 9922833089, 156975157885, 551852740992, 551852740992, 156975157885, 9922833089, 55125128, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(7*n-13)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) + 7*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 + (7*n - 13)*x)*P[n - 1, x] + (7*x - 7*x^2)*D[P[n - 1, x], x] + 7*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 + (7*n - 13)*x)*v[i - 1] + (7*x - 7*x^2)*deriv(v[i - 1], x) + 7*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.

A144445: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       23,          1;
   4: 1,      206,        206,            1;
   5: 1,     1677,       6341,         1677,            1;
   6: 1,    13452,     133451,       133451,        13452,            1;
   7: 1,   107659,    2403612,      5916231,      2403612,       107659,            1;
   8: 1,   861322,   40024068,    200795987,    200795987,     40024068,       861322,          1;
   9: 1,  6890633,  638151479,   5875203446,  11687580863,   5875203446,    638151479,    6890633,        1;
  10: 1, 55125128, 9922833089, 156975157885, 551852740992, 551852740992, 156975157885, 9922833089, 55125128, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(7*n-13)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) + 7*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 + (7*n - 13)*x)*P[n - 1, x] + (7*x - 7*x^2)*D[P[n - 1, x], x] + 7*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 + (7*n - 13)*x)*v[i - 1] + (7*x - 7*x^2)*deriv(v[i - 1], x) + 7*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-789cbd055a27828e
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-789cbd055a27828e
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-789cbd055a27828e
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-789cbd055a27828e

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-789cbd055a27828e root: theorem check

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

Used in 1 proof step

e-fe58044f3c49833e gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*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