A144442: Triangle read by rows: T(n, k) = (5n-5k+1)T(n-1, k-1) +(5k-4)T(n-1, k) + 5T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

A144442 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 A144442

\(P_{n}(x) = \left(1 - 4\,x + 5\,n x\right)\,P_{n-1}(x) + \left(5\,x - 5\,x^{2}\right)\,P_{n-1}'(x) + \left(5\,x\right)\,P_{n-2}(x)\)

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

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

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	17	1													19
3	1	118	118	1												238
4	1	729	2681	729	1											4141
5	1	4400	41745	41745	4400	1										92292
6	1	26431	555240	1349245	555240	26431	1									2512589
7	1	158622	6816846	33456685	33456685	6816846	158622	1								80864308
8	1	951773	80034743	715321156	1411926995	715321156	80034743	951773	1							3004542341
9	1	5710684	915439111	13960299759	48416100376	48416100376	13960299759	915439111	5710684	1						126595099862
10	1	34264155	10308727130	256720777855	1453084006205	2524696854527	1453084006205	256720777855	10308727130	34264155	1					5964992405219
11	1	205584986	115000702980	4534767453565	39826513631880	110929802911937	110929802911937	39826513631880	4534767453565	115000702980	205584986	1				310812580570698
12	1	1233509977	1275663887841	77897855229770	1023565855754920	4325194786489067	6890271264812729	4325194786489067	1023565855754920	77897855229770	1275663887841	1233509977	1			17746142054555881
13	1	7401059928	14102407249893	1311999545471111	25100858148690565	154620397102826862	369860070537360696	369860070537360696	154620397102826862	25100858148690565	1311999545471111	14102407249893	7401059928	1		1101814870285318112
14	1	44406359639	155584111954316	21788105852970989	594419487217677376	5179887628792039002	17826724441806528253	26664376435014033757	17826724441806528253	5179887628792039002	594419487217677376	21788105852970989	155584111954316	44406359639	1	73910327019389092909

Generated b-files: A144442, A144442 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-aac530d688cc3150 · revision 1 · JSON
Definition source: oeis_catalog:A144442
Fingerprint: f6-568f7034a48e0079

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 ?

A144442: 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 ?
A144442: 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,      17,         1;
  3: 1,     118,       118,           1;
  4: 1,     729,      2681,         729,           1;
  5: 1,    4400,     41745,       41745,        4400,           1;
  6: 1,   26431,    555240,     1349245,      555240,       26431,           1;
  7: 1,  158622,   6816846,    33456685,    33456685,     6816846,      158622,         1;
  8: 1,  951773,  80034743,   715321156,  1411926995,   715321156,    80034743,    951773,       1;
  9: 1, 5710684, 915439111, 13960299759, 48416100376, 48416100376, 13960299759, 915439111, 5710684, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(5*n-4)*x)*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) + 5*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 + (5*n - 4)*x)*P[n - 1, x] + (5*x - 5*x^2)*D[P[n - 1, x], x] + 5*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 + (5*n - 4)*x)*v[i - 1] + (5*x - 5*x^2)*deriv(v[i - 1], x) + 5*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.

A144442: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,       1;
   3: 1,      17,         1;
   4: 1,     118,       118,           1;
   5: 1,     729,      2681,         729,           1;
   6: 1,    4400,     41745,       41745,        4400,           1;
   7: 1,   26431,    555240,     1349245,      555240,       26431,           1;
   8: 1,  158622,   6816846,    33456685,    33456685,     6816846,      158622,         1;
   9: 1,  951773,  80034743,   715321156,  1411926995,   715321156,    80034743,    951773,       1;
  10: 1, 5710684, 915439111, 13960299759, 48416100376, 48416100376, 13960299759, 915439111, 5710684, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(5*n-9)*x)*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) + 5*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 + (5*n - 9)*x)*P[n - 1, x] + (5*x - 5*x^2)*D[P[n - 1, x], x] + 5*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 + (5*n - 9)*x)*v[i - 1] + (5*x - 5*x^2)*deriv(v[i - 1], x) + 5*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.

A144442: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,       1;
   3: 1,      17,         1;
   4: 1,     118,       118,           1;
   5: 1,     729,      2681,         729,           1;
   6: 1,    4400,     41745,       41745,        4400,           1;
   7: 1,   26431,    555240,     1349245,      555240,       26431,           1;
   8: 1,  158622,   6816846,    33456685,    33456685,     6816846,      158622,         1;
   9: 1,  951773,  80034743,   715321156,  1411926995,   715321156,    80034743,    951773,       1;
  10: 1, 5710684, 915439111, 13960299759, 48416100376, 48416100376, 13960299759, 915439111, 5710684, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(5*n-9)*x)*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) + 5*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 + (5*n - 9)*x)*P[n - 1, x] + (5*x - 5*x^2)*D[P[n - 1, x], x] + 5*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 + (5*n - 9)*x)*v[i - 1] + (5*x - 5*x^2)*deriv(v[i - 1], x) + 5*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-0086426c09dadf26
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-0086426c09dadf26
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-0086426c09dadf26
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-0086426c09dadf26

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-0086426c09dadf26 root: theorem check

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

Used in 1 proof step

e-a34157123148cce9 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n…

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