A113413 → reverse

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(1 + n + 2\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 3\,x^{2} + 2\,x^{3} - n x - 6\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(x^{2} + 6\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)

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

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	2														3
2	1	4	2													7
3	1	6	8	2												17
4	1	8	18	12	2											41
5	1	10	32	38	16	2										99
6	1	12	50	88	66	20	2									239
7	1	14	72	170	192	102	24	2								577
8	1	16	98	292	450	360	146	28	2							1393
9	1	18	128	462	912	1002	608	198	32	2						3363
10	1	20	162	688	1666	2364	1970	952	258	36	2					8119
11	1	22	200	978	2816	4942	5336	3530	1408	326	40	2				19601
12	1	24	242	1340	4482	9424	12642	10836	5890	1992	402	44	2			47321
13	1	26	288	1782	6800	16722	27008	28814	20256	9290	2720	486	48	2		114243
14	1	28	338	2312	9922	28004	53154	68464	59906	35436	14002	3608	578	52	2	275807

Generated b-files: A035607, A078057 (row sums), A113413 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-170d9d09425f5556 · revision 1 · JSON
Definition source: derived:reversal
Fingerprint: f6-f7ea49607396af2f

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: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A035607: triangle, 71 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A078057: 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 ?
A113413: reversed triangle, 69 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A001333: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,  2;
  2: 1,  4,   2;
  3: 1,  6,   8,   2;
  4: 1,  8,  18,  12,   2;
  5: 1, 10,  32,  38,  16,    2;
  6: 1, 12,  50,  88,  66,   20,   2;
  7: 1, 14,  72, 170, 192,  102,  24,   2;
  8: 1, 16,  98, 292, 450,  360, 146,  28,  2;
  9: 1, 18, 128, 462, 912, 1002, 608, 198, 32, 2;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = (n+1+(2*n^2+2*n)*x)*P(n-1,x) - ((n+1)*x+(6*n-3)*x^2+(n-2)*x^3)*d/dx P(n-1,x) + (x^2+6*x^3+x^4)*d^2/dx^2 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[Together[((n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x] - ((n + 1)*x + (6*n - 3)*x^2 + (n - 2)*x^3)*D[P[n - 1, x], x] + (x^2 + 6*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n + 1)]];
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] = ((n + 1 + (2*n^2 + 2*n)*x)*v[i - 1] - ((n + 1)*x + (6*n - 3)*x^2 + (n - 2)*x^3)*deriv(v[i - 1], x) + (x^2 + 6*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(n + 1)); 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.

A035607: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  4,   2;
  3: 1,  6,   8,   2;
  4: 1,  8,  18,  12,   2;
  5: 1, 10,  32,  38,  16,    2;
  6: 1, 12,  50,  88,  66,   20,   2;
  7: 1, 14,  72, 170, 192,  102,  24,   2;
  8: 1, 16,  98, 292, 450,  360, 146,  28,  2;
  9: 1, 18, 128, 462, 912, 1002, 608, 198, 32, 2;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = (n+1+(2*n^2+2*n)*x)*P(n-1,x) - ((n+1)*x+(6*n-3)*x^2+(n-2)*x^3)*d/dx P(n-1,x) + (x^2+6*x^3+x^4)*d^2/dx^2 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[Together[((n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x] - ((n + 1)*x + (6*n - 3)*x^2 + (n - 2)*x^3)*D[P[n - 1, x], x] + (x^2 + 6*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n + 1)]];
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] = ((n + 1 + (2*n^2 + 2*n)*x)*v[i - 1] - ((n + 1)*x + (6*n - 3)*x^2 + (n - 2)*x^3)*deriv(v[i - 1], x) + (x^2 + 6*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(n + 1)); 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.

A113413: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 2,  1;
  2: 2,  4,   1;
  3: 2,  8,   6,   1;
  4: 2, 12,  18,   8,    1;
  5: 2, 16,  38,  32,   10,   1;
  6: 2, 20,  66,  88,   50,  12,   1;
  7: 2, 24, 102, 192,  170,  72,  14,   1;
  8: 2, 28, 146, 360,  450, 292,  98,  16,  1;
  9: 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (n+1+(2*n^2+2*n)*x)*P(n-1,x) - ((n+1)*x+(6*n-3)*x^2+(n-2)*x^3)*d/dx P(n-1,x) + (x^2+6*x^3+x^4)*d^2/dx^2 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[Together[((n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x] - ((n + 1)*x + (6*n - 3)*x^2 + (n - 2)*x^3)*D[P[n - 1, x], x] + (x^2 + 6*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n + 1)]];
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] = ((n + 1 + (2*n^2 + 2*n)*x)*v[i - 1] - ((n + 1)*x + (6*n - 3)*x^2 + (n - 2)*x^3)*deriv(v[i - 1], x) + (x^2 + 6*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(n + 1)); 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-de1b791dfb436520
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-de1b791dfb436520
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-de1b791dfb436520

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

preserver_image reversal, shift 0 e-de1b791dfb436520

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): A Riordan array of coordination sequences.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-7fe1799d89540b32 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

Same fingerprint (6)

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