A257626: 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) = 3*x + 6.

A257626 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 A257626

\(P_{n}(x) = \left(6 + 3\,x + 3\,n x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	6														12
2	36	108	36													180
3	216	1404	1404	216												3240
4	1296	15876	33696	15876	1296											68040
5	7776	166212	642492	642492	166212	7776										1632960
6	46656	1659204	10701720	19274760	10701720	1659204	46656									44089920
7	279936	16052580	163263924	481752360	481752360	163263924	16052580	279936								1322697600
8	1679616	152031492	2344429008	10654827804	17343084960	10654827804	2344429008	152031492	1679616							43649020800
9	10077696	1418671908	32237998380	216088713252	535926913164	535926913164	216088713252	32237998380	1418671908	10077696						1571364748800
10	60466176	13100611140	429416137800	4111756655040	14832813555000	22508930352888	14832813555000	4111756655040	429416137800	13100611140	60466176					61283225203200
11	362797056	120082282596	5585313821220	74558833959600	378008073676080	828675062730648	828675062730648	378008073676080	74558833959600	5585313821220	120082282596	362797056				2573895458534400
12	2176782336	1094889628548	71346728028096	1302697865494260	9040910344957440	27608394306597768	39776403011071104	27608394306597768	9040910344957440	1302697865494260	71346728028096	1094889628548	2176782336			115825295634048000
13	13060694016	9945431515044	898861431850524	22108950191425356	205725415770544500	851003590787276328	1700060318543846232	1700060318543846232	851003590787276328	205725415770544500	22108950191425356	898861431850524	9945431515044	13060694016		5559614190434304000
14	78364164096	90096614866116	11204045305838136	366689848713550776	4498979690761113816	24660014126960771388	66331555368670599408	91803257201367696528	66331555368670599408	24660014126960771388	4498979690761113816	366689848713550776	11204045305838136	90096614866116	78364164096	283540323712149504000

Generated b-files: A051609 (row sums), A257626, A257626 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-61cf3a32da4a0e22 · revision 1 · JSON
Definition source: oeis_catalog:A257626
Fingerprint: f6-9f31c05ba8767dc5

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 ?

A257626: reversed triangle, 36 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A051609: 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 ?
A257626: triangle, 36 terms agree, 8 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:        6,          6;
  2:       36,        108,          36;
  3:      216,       1404,        1404,          216;
  4:     1296,      15876,       33696,        15876,         1296;
  5:     7776,     166212,      642492,       642492,       166212,         7776;
  6:    46656,    1659204,    10701720,     19274760,     10701720,      1659204,        46656;
  7:   279936,   16052580,   163263924,    481752360,    481752360,    163263924,     16052580,      279936;
  8:  1679616,  152031492,  2344429008,  10654827804,  17343084960,  10654827804,   2344429008,   152031492,    1679616;
  9: 10077696, 1418671908, 32237998380, 216088713252, 535926913164, 535926913164, 216088713252, 32237998380, 1418671908, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (6+(3*n+3)*x)*P(n-1,x) + (3*x-3*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[(6 + (3*n + 3)*x)*P[n - 1, x] + (3*x - 3*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] = (6 + (3*n + 3)*x)*v[i - 1] + (3*x - 3*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.

A257626: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:        1;
  1:        6,          6;
  2:       36,        108,          36;
  3:      216,       1404,        1404,          216;
  4:     1296,      15876,       33696,        15876,         1296;
  5:     7776,     166212,      642492,       642492,       166212,         7776;
  6:    46656,    1659204,    10701720,     19274760,     10701720,      1659204,        46656;
  7:   279936,   16052580,   163263924,    481752360,    481752360,    163263924,     16052580,      279936;
  8:  1679616,  152031492,  2344429008,  10654827804,  17343084960,  10654827804,   2344429008,   152031492,    1679616;
  9: 10077696, 1418671908, 32237998380, 216088713252, 535926913164, 535926913164, 216088713252, 32237998380, 1418671908, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (6+(3*n+3)*x)*P(n-1,x) + (3*x-3*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[(6 + (3*n + 3)*x)*P[n - 1, x] + (3*x - 3*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] = (6 + (3*n + 3)*x)*v[i - 1] + (3*x - 3*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.

A257626: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:        1;
  1:        6,          6;
  2:       36,        108,          36;
  3:      216,       1404,        1404,          216;
  4:     1296,      15876,       33696,        15876,         1296;
  5:     7776,     166212,      642492,       642492,       166212,         7776;
  6:    46656,    1659204,    10701720,     19274760,     10701720,      1659204,        46656;
  7:   279936,   16052580,   163263924,    481752360,    481752360,    163263924,     16052580,      279936;
  8:  1679616,  152031492,  2344429008,  10654827804,  17343084960,  10654827804,   2344429008,   152031492,    1679616;
  9: 10077696, 1418671908, 32237998380, 216088713252, 535926913164, 535926913164, 216088713252, 32237998380, 1418671908, 10077696;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+(3*n+3)*x)*P(n-1,x) + (3*x-3*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[(6 + (3*n + 3)*x)*P[n - 1, x] + (3*x - 3*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] = (6 + (3*n + 3)*x)*v[i - 1] + (3*x - 3*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-5ba7ec4213cfb517
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-5ba7ec4213cfb517
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-5ba7ec4213cfb517
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-5ba7ec4213cfb517

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-5ba7ec4213cfb517 root: theorem check

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

Used in 3 proof steps

e-34a6daf3526c7cea 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-0e437c53d6b0d63a 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-0e437c53d6b0d63a 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 ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

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