A257607: 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) = x + 5.

A257607 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 A257607

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	5														10
2	25	60	25													110
3	125	535	535	125												1320
4	625	4210	7490	4210	625											17160
5	3125	30885	86110	86110	30885	3125										240240
6	15625	216560	880735	1377760	880735	216560	15625									3603600
7	78125	1471235	8330745	18948695	18948695	8330745	1471235	78125								57657600
8	390625	9764910	74498800	234897010	341076510	234897010	74498800	9764910	390625							980179200
9	1953125	63667585	638670520	2698662880	5418658690	5418658690	2698662880	638670520	63667585	1953125						17643225600
10	9765625	409349260	5298372245	29253349280	78453219890	108373173800	78453219890	29253349280	5298372245	409349260	9765625					335221286400
11	48828125	2602579935	42819495355	302905633425	1057119170370	1946717156790	1946717156790	1057119170370	302905633425	42819495355	2602579935	48828125				6704425728000
12	244140625	16396729610	338775166510	3022718002370	13451845767855	32152601612340	42827777449380	32152601612340	13451845767855	3022718002370	338775166510	16396729610	244140625			140792940288000
13	1220703125	102530768285	2633773839330	29263371516610	163384663943875	496400011105515	856936771291260	856936771291260	496400011105515	163384663943875	29263371516610	2633773839330	102530768285	1220703125		3097444686336000
14	6103515625	637157265960	20179439936155	276247353562160	1909412548244025	7251385406269400	15879504628575555	20566482510990240	15879504628575555	7251385406269400	1909412548244025	276247353562160	20179439936155	637157265960	6103515625	71241227785728000

Generated b-files: A049398 (row sums), A257607, A257607 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-8919a5e7419ba068 · revision 1 · JSON
Definition source: oeis_catalog:A257607
Fingerprint: f6-c2587be023370403

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 ?

A049398: 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 ?
A257607: 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 = 0 ?
A257607: 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 = 0 ?

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:       5,        5;
  2:      25,       60,        25;
  3:     125,      535,       535,        125;
  4:     625,     4210,      7490,       4210,        625;
  5:    3125,    30885,     86110,      86110,      30885,       3125;
  6:   15625,   216560,    880735,    1377760,     880735,     216560,      15625;
  7:   78125,  1471235,   8330745,   18948695,   18948695,    8330745,    1471235,     78125;
  8:  390625,  9764910,  74498800,  234897010,  341076510,  234897010,   74498800,   9764910,   390625;
  9: 1953125, 63667585, 638670520, 2698662880, 5418658690, 5418658690, 2698662880, 638670520, 63667585, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (5+(n+4)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + (n + 4)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + (n + 4)*x)*v[i - 1] + (x - 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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A257607: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:       1;
  1:       5,        5;
  2:      25,       60,        25;
  3:     125,      535,       535,        125;
  4:     625,     4210,      7490,       4210,        625;
  5:    3125,    30885,     86110,      86110,      30885,       3125;
  6:   15625,   216560,    880735,    1377760,     880735,     216560,      15625;
  7:   78125,  1471235,   8330745,   18948695,   18948695,    8330745,    1471235,     78125;
  8:  390625,  9764910,  74498800,  234897010,  341076510,  234897010,   74498800,   9764910,   390625;
  9: 1953125, 63667585, 638670520, 2698662880, 5418658690, 5418658690, 2698662880, 638670520, 63667585, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (5+(n+4)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + (n + 4)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + (n + 4)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A257607: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:       1;
  1:       5,        5;
  2:      25,       60,        25;
  3:     125,      535,       535,        125;
  4:     625,     4210,      7490,       4210,        625;
  5:    3125,    30885,     86110,      86110,      30885,       3125;
  6:   15625,   216560,    880735,    1377760,     880735,     216560,      15625;
  7:   78125,  1471235,   8330745,   18948695,   18948695,    8330745,    1471235,     78125;
  8:  390625,  9764910,  74498800,  234897010,  341076510,  234897010,   74498800,   9764910,   390625;
  9: 1953125, 63667585, 638670520, 2698662880, 5418658690, 5418658690, 2698662880, 638670520, 63667585, 1953125;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+(n+4)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + (n + 4)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + (n + 4)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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-5eeeda74dfe3072e
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-5eeeda74dfe3072e
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-5eeeda74dfe3072e
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-5eeeda74dfe3072e

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-5eeeda74dfe3072e root: theorem check

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

Used in 3 proof steps

e-a5ce40235f8a7dd2 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f…
e-e045f1dbdfaaf32d obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1…
e-e045f1dbdfaaf32d obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1…

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