A257608: 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) = 9*x + 1.

A257608 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 A257608

\(P_{n}(x) = \left(1 - 8\,x + 9\,n x\right)\,P_{n-1}(x) + \left(9\,x - 9\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	1	20	1													22
3	1	219	219	1												440
4	1	2218	8322	2218	1											12760
5	1	22217	220222	220222	22217	1										484880
6	1	222216	5006247	12332432	5006247	222216	1									22789360
7	1	2222215	105340629	530539235	530539235	105340629	2222215	1								1276204160
8	1	22222214	2123693776	19700767514	39259903390	19700767514	2123693776	22222214	1							82953270400
9	1	222222213	41772403440	668424648072	2358851731074	2358851731074	668424648072	41772403440	222222213	1						6138542009600
10	1	2222222212	809897886909	21389323966176	124040869693698	217014359258808	124040869693698	21389323966176	809897886909	2222222212	1					509498986796800
11	1	22222222211	15570282072655	658023616797285	5958428912502090	16804908359058558	16804908359058558	5958428912502090	658023616797285	15570282072655	22222222211	1				46873906785305600
12	1	222222222210	297857581601646	19701424400281690	268497593788779135	1154365234916827428	1848539919496441380	1154365234916827428	268497593788779135	19701424400281690	297857581601646	222222222210	1			4734264585315865600
13	1	2222222222209	5681516272652274	578744923133637106	11549927771007926575	72701125152754938543	175549070606981231292	175549070606981231292	72701125152754938543	11549927771007926575	578744923133637106	5681516272652274	2222222222209	1		520769104384745216000
14	1	22222222222208	108191031402613987	16773009475007066368	480013115532454259921	4291345834249377152128	14962381019535078234699	22470281037693597605376	14962381019535078234699	4291345834249377152128	480013115532454259921	16773009475007066368	108191031402613987	22222222222208	1	61971523421784680704000

Generated b-files: A084949 (row sums), A257608, A257608 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-dcd2d11144e6ea21 · revision 1 · JSON
Definition source: oeis_catalog:A257608
Fingerprint: f6-bb7d69d9a9c0913d

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 ?

A257608: 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 ?
A084949: 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 ?
A257608: 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: 1,         1;
  2: 1,        20,           1;
  3: 1,       219,         219,            1;
  4: 1,      2218,        8322,         2218,             1;
  5: 1,     22217,      220222,       220222,         22217,             1;
  6: 1,    222216,     5006247,     12332432,       5006247,        222216,            1;
  7: 1,   2222215,   105340629,    530539235,     530539235,     105340629,      2222215,           1;
  8: 1,  22222214,  2123693776,  19700767514,   39259903390,   19700767514,   2123693776,    22222214,         1;
  9: 1, 222222213, 41772403440, 668424648072, 2358851731074, 2358851731074, 668424648072, 41772403440, 222222213, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(9*n-8)*x)*P(n-1,x) + (9*x-9*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[(1 + (9*n - 8)*x)*P[n - 1, x] + (9*x - 9*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] = (1 + (9*n - 8)*x)*v[i - 1] + (9*x - 9*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.

A257608: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,         1;
  2: 1,        20,           1;
  3: 1,       219,         219,            1;
  4: 1,      2218,        8322,         2218,             1;
  5: 1,     22217,      220222,       220222,         22217,             1;
  6: 1,    222216,     5006247,     12332432,       5006247,        222216,            1;
  7: 1,   2222215,   105340629,    530539235,     530539235,     105340629,      2222215,           1;
  8: 1,  22222214,  2123693776,  19700767514,   39259903390,   19700767514,   2123693776,    22222214,         1;
  9: 1, 222222213, 41772403440, 668424648072, 2358851731074, 2358851731074, 668424648072, 41772403440, 222222213, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(9*n-8)*x)*P(n-1,x) + (9*x-9*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[(1 + (9*n - 8)*x)*P[n - 1, x] + (9*x - 9*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] = (1 + (9*n - 8)*x)*v[i - 1] + (9*x - 9*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.

A257608: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,         1;
  2: 1,        20,           1;
  3: 1,       219,         219,            1;
  4: 1,      2218,        8322,         2218,             1;
  5: 1,     22217,      220222,       220222,         22217,             1;
  6: 1,    222216,     5006247,     12332432,       5006247,        222216,            1;
  7: 1,   2222215,   105340629,    530539235,     530539235,     105340629,      2222215,           1;
  8: 1,  22222214,  2123693776,  19700767514,   39259903390,   19700767514,   2123693776,    22222214,         1;
  9: 1, 222222213, 41772403440, 668424648072, 2358851731074, 2358851731074, 668424648072, 41772403440, 222222213, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(9*n-8)*x)*P(n-1,x) + (9*x-9*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[(1 + (9*n - 8)*x)*P[n - 1, x] + (9*x - 9*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] = (1 + (9*n - 8)*x)*v[i - 1] + (9*x - 9*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-e4e9153fc21b93ec
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-e4e9153fc21b93ec
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-e4e9153fc21b93ec
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-e4e9153fc21b93ec

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-e4e9153fc21b93ec root: theorem check

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

Used in 3 proof steps

e-96a51d1c5f8c0af4 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-c1abd10180bf6e94 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-c1abd10180bf6e94 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