A257616: 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) = 6*x + 2.

A257616 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 A257616

\(P_{n}(x) = \left(2 - 4\,x + 6\,n x\right)\,P_{n-1}(x) + \left(6\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	4	32	4													40
3	8	312	312	8												640
4	16	2656	8736	2656	16											14080
5	32	21664	175424	175424	21664	32										394240
6	64	174336	3019200	7016960	3019200	174336	64									13404160
7	128	1397120	47847552	218838400	218838400	47847552	1397120	128								536166400
8	256	11182592	722956288	5907889664	11379596800	5907889664	722956288	11182592	256							24663654400
9	512	89473536	10613422080	145630132224	484921986048	484921986048	145630132224	10613422080	89473536	512						1282510028800
10	1024	715816960	153061585920	3379593216000	18141916661760	31035007107072	18141916661760	3379593216000	153061585920	715816960	1024					74385581670400
11	2048	5726599168	2182947952640	75244943616000	620391934709760	1682513060573184	1682513060573184	620391934709760	75244943616000	2182947952640	5726599168	2048				4760677226905600
12	4096	45812932608	30916320485376	1627143957667840	19892437483253760	81137663065571328	127870992603561984	81137663065571328	19892437483253760	1627143957667840	30916320485376	45812932608	4096			333247405883392000
13	8192	366503763968	435943766212608	34459691023450112	608323436193996800	3591027092260970496	8429154893820493824	8429154893820493824	3591027092260970496	608323436193996800	34459691023450112	435943766212608	366503763968	8192		25326802847137792000
14	16384	2932030767104	6130334005510144	718837996571459584	17952910184497823744	148978979379214876672	499859240578227290112	741765630656203456512	499859240578227290112	148978979379214876672	17952910184497823744	718837996571459584	6130334005510144	2932030767104	16384	2076797833465298944000

Generated b-files: A049308 (row sums), A257616, A257616 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-c060a08f0b5770ad · revision 1 · JSON
Definition source: oeis_catalog:A257616
Fingerprint: f6-257640823bee26bf

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 ?

A049308: 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 ?
A257616: 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 ?
A257616: 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:   2,        2;
  2:   4,       32,           4;
  3:   8,      312,         312,            8;
  4:  16,     2656,        8736,         2656,           16;
  5:  32,    21664,      175424,       175424,        21664,           32;
  6:  64,   174336,     3019200,      7016960,      3019200,       174336,           64;
  7: 128,  1397120,    47847552,    218838400,    218838400,     47847552,      1397120,         128;
  8: 256, 11182592,   722956288,   5907889664,  11379596800,   5907889664,    722956288,    11182592,      256;
  9: 512, 89473536, 10613422080, 145630132224, 484921986048, 484921986048, 145630132224, 10613422080, 89473536, 512;

%F Formula

The row polynomials satisfy P(n,x) = (2+(6*n-4)*x)*P(n-1,x) + (6*x-6*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[(2 + (6*n - 4)*x)*P[n - 1, x] + (6*x - 6*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] = (2 + (6*n - 4)*x)*v[i - 1] + (6*x - 6*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.

A257616: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:   1;
  1:   2,        2;
  2:   4,       32,           4;
  3:   8,      312,         312,            8;
  4:  16,     2656,        8736,         2656,           16;
  5:  32,    21664,      175424,       175424,        21664,           32;
  6:  64,   174336,     3019200,      7016960,      3019200,       174336,           64;
  7: 128,  1397120,    47847552,    218838400,    218838400,     47847552,      1397120,         128;
  8: 256, 11182592,   722956288,   5907889664,  11379596800,   5907889664,    722956288,    11182592,      256;
  9: 512, 89473536, 10613422080, 145630132224, 484921986048, 484921986048, 145630132224, 10613422080, 89473536, 512;

%F Formula

The row polynomials satisfy P(n,x) = (2+(6*n-4)*x)*P(n-1,x) + (6*x-6*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[(2 + (6*n - 4)*x)*P[n - 1, x] + (6*x - 6*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] = (2 + (6*n - 4)*x)*v[i - 1] + (6*x - 6*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.

A257616: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:   1;
  1:   2,        2;
  2:   4,       32,           4;
  3:   8,      312,         312,            8;
  4:  16,     2656,        8736,         2656,           16;
  5:  32,    21664,      175424,       175424,        21664,           32;
  6:  64,   174336,     3019200,      7016960,      3019200,       174336,           64;
  7: 128,  1397120,    47847552,    218838400,    218838400,     47847552,      1397120,         128;
  8: 256, 11182592,   722956288,   5907889664,  11379596800,   5907889664,    722956288,    11182592,      256;
  9: 512, 89473536, 10613422080, 145630132224, 484921986048, 484921986048, 145630132224, 10613422080, 89473536, 512;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+(6*n-4)*x)*P(n-1,x) + (6*x-6*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[(2 + (6*n - 4)*x)*P[n - 1, x] + (6*x - 6*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] = (2 + (6*n - 4)*x)*v[i - 1] + (6*x - 6*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-d30c56e449c0455e
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-d30c56e449c0455e
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-d30c56e449c0455e
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-d30c56e449c0455e

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-d30c56e449c0455e root: theorem check

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

Used in 3 proof steps

e-adb175137addd81e 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-4b3d723d1668996c 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-4b3d723d1668996c 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 (1)

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

A142458: Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(…

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