A253284 → basis A013609

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(-3 - 4\,x + 6\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	4							8
2	36	84	48						168
3	528	1968	2400	960					5856
4	10800	55440	105120	87360	26880				285600
5	283680	1856160	4798080	6128640	3870720	967680			17904960
6	9102240	72404640	237484800	411264000	396748800	202245120	42577920		1371827520
7	345058560	3232131840	12859015680	28174003200	36723456000	28484628480	12177285120	2214051840	124209630720

All 15 rows

Technical details

Node id: n-3e752e21877e7b02 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-8419344160e8e665

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:            1;
  1:            4,             4;
  2:           36,            84,             48;
  3:          528,          1968,           2400,             960;
  4:        10800,         55440,         105120,           87360,           26880;
  5:       283680,       1856160,        4798080,         6128640,         3870720,          967680;
  6:      9102240,      72404640,      237484800,       411264000,       396748800,       202245120,        42577920;
  7:    345058560,    3232131840,    12859015680,     28174003200,     36723456000,     28484628480,     12177285120,      2214051840;
  8:  15090727680,  162674023680,   761184345600,   2019630412800,   3323951769600,   3475507875840,   2255011799040,    830269440000,   132843110400;
  9: 747888422400, 9118920384000, 49072117248000, 152986724352000, 304542830592000, 401485090406400, 350572968345600, 195545058508800, 63233320550400, 9033331507200;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-2*n)*P(n-2,x) + (6*n-3+(8*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n)*P[n - 2, x] + (6*n - 3 + (8*n - 4)*x)*P[n - 1, x]];
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] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n)*v[i - 2] + (6*n - 3 + (8*n - 4)*x)*v[i - 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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 read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

A088617 → basis A013609

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.