A257612: 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) = 4*x + 2.

A257612 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 A257612

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

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

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	24	4													32
3	8	184	184	8												384
4	16	1216	3680	1216	16											6144
5	32	7584	53824	53824	7584	32										122880
6	64	46208	674752	1507072	674752	46208	64									2949120
7	128	278912	7764096	33244544	33244544	7764096	278912	128								82575360
8	256	1677312	84892672	636233728	1196803584	636233728	84892672	1677312	256							2642411520
9	512	10072576	899246080	11114481664	35539606528	35539606528	11114481664	899246080	10072576	512						95126814720
10	1024	60454912	9334928384	182580125696	928689440768	1563742687232	928689440768	182580125696	9334928384	60454912	1024					3805072588800
11	2048	362772480	95646570496	2873509324800	22193813704704	58548264579072	58548264579072	22193813704704	2873509324800	95646570496	362772480	2048				167423193907200
12	4096	2176729088	971702149120	43863700226048	497187963727872	1953876231880704	3044509758111744	1953876231880704	497187963727872	43863700226048	971702149120	2176729088	4096			8036313307545600
13	8192	13060579328	9817151029248	654903293427712	10616203955691520	59889667868123136	137773540667326464	137773540667326464	59889667868123136	10616203955691520	654903293427712	9817151029248	13060579328	8192		417888291992371200
14	16384	78363918336	98824539258880	9620235055333376	218597609526411264	1720988443414986752	5618360764866674688	8266412440039587840	5618360764866674688	1720988443414986752	218597609526411264	9620235055333376	98824539258880	78363918336	16384	23401744351572787200

Generated b-files: A047053 (row sums), A257612, A257612 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-2c43e713cd32b015 · revision 1 · JSON
Definition source: oeis_catalog:A257612
Fingerprint: f6-57e3a3defd926b1b

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 ?

A257612: 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 ?
A047053: 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 ?
A257612: 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,       24,         4;
  3:   8,      184,       184,           8;
  4:  16,     1216,      3680,        1216,          16;
  5:  32,     7584,     53824,       53824,        7584,          32;
  6:  64,    46208,    674752,     1507072,      674752,       46208,          64;
  7: 128,   278912,   7764096,    33244544,    33244544,     7764096,      278912,       128;
  8: 256,  1677312,  84892672,   636233728,  1196803584,   636233728,    84892672,   1677312,      256;
  9: 512, 10072576, 899246080, 11114481664, 35539606528, 35539606528, 11114481664, 899246080, 10072576, 512;

%F Formula

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

A257612: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:   1;
  1:   2,        2;
  2:   4,       24,         4;
  3:   8,      184,       184,           8;
  4:  16,     1216,      3680,        1216,          16;
  5:  32,     7584,     53824,       53824,        7584,          32;
  6:  64,    46208,    674752,     1507072,      674752,       46208,          64;
  7: 128,   278912,   7764096,    33244544,    33244544,     7764096,      278912,       128;
  8: 256,  1677312,  84892672,   636233728,  1196803584,   636233728,    84892672,   1677312,      256;
  9: 512, 10072576, 899246080, 11114481664, 35539606528, 35539606528, 11114481664, 899246080, 10072576, 512;

%F Formula

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

A257612: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:   1;
  1:   2,        2;
  2:   4,       24,         4;
  3:   8,      184,       184,           8;
  4:  16,     1216,      3680,        1216,          16;
  5:  32,     7584,     53824,       53824,        7584,          32;
  6:  64,    46208,    674752,     1507072,      674752,       46208,          64;
  7: 128,   278912,   7764096,    33244544,    33244544,     7764096,      278912,       128;
  8: 256,  1677312,  84892672,   636233728,  1196803584,   636233728,    84892672,   1677312,      256;
  9: 512, 10072576, 899246080, 11114481664, 35539606528, 35539606528, 11114481664, 899246080, 10072576, 512;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+(4*n-2)*x)*P(n-1,x) + (4*x-4*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 + (4*n - 2)*x)*P[n - 1, x] + (4*x - 4*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 + (4*n - 2)*x)*v[i - 1] + (4*x - 4*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-1498aa51594830e4
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-1498aa51594830e4
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-1498aa51594830e4
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-1498aa51594830e4

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-1498aa51594830e4 root: theorem check

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

Used in 3 proof steps

e-b43c1d901148031a 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-697623ff3ff432a2 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-697623ff3ff432a2 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 (3)

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