A185263 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	1							3
3	6	5							11
4	24	31	8						63
5	120	203	84						407
6	720	1548	1009	180					3457
7	5040	12996	11001	3044					32081
8	40320	122616	132549	58316	8064				361865
9	362880	1264824	1634589	925892	193248				4381433
10	3628800	14341680	21898494	15891449	5310636	604800			61675859
11	39916800	175902480	307354554	265725109	113413196	19056960			921369099
12	479001600	2336074560	4631958288	4737685107	2597215396	702844416	68428800		15553208167
13	6227020800	33242978880	73427992272	85795623543	55843755992	19159839472	2699672832		276396883791
14	87178291200	506655717120	1238984000928	1643770909176	1267189475329	559690673016	127662818256	10897286400	5442029171425

Technical details

Node id: n-bc0f8670bc032a43 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-853f937479c1e1fc

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      1;
  2:      2,       1;
  3:      6,       5;
  4:     24,      31,       8;
  5:    120,     203,      84;
  6:    720,    1548,    1009,    180;
  7:   5040,   12996,   11001,   3044;
  8:  40320,  122616,  132549,  58316,   8064;
  9: 362880, 1264824, 1634589, 925892, 193248;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (n^3-n^2-n+1+(n^3-n^2)*x)*P(n-2,x) + (2*n+1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 - n + 1 + (n^3 - n^2)*x)*P[n - 2, x] + (2*n + 1)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - n^2 - n + 1 + (n^3 - n^2)*x)*v[i - 2] + (2*n + 1)*v[i - 1])/(n + 2)); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t)…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t)…?
  - ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A094638 → Veronese 2 (j=1) → x ↦ x+1

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