A024462 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + x\)

\(P_{2}(x) = 4 + 5\,x + x^{2}\)

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	1														3
2	4	5	1													10
3	16	47	25	3												91
4	64	377	391	114	9											955
5	256	2831	5041	2655	486	27										11296
6	1024	20585	58903	49638	15741	1971	81									147943
7	4096	147167	650785	822003	400770	84672	7695	243								2117431
8	16384	1042457	6949351	12638394	8878329	2811078	423306	29160	729							32789188
9	65536	7346351	72620881	185147175	179968446	80045469	17745966	1998918	107892	2187						545048821
10	262144	51621065	748247863	2624775918	3434936661	2060769249	630547659	103210848	9017730	391473	6561					9663787171
11	1048576	362133887	7637341825	36366830523	62833314330	49459425714	20054356245	4471914177	562128984	39188853	1397493	19683				181789100290
12	4194304	2538082937	77459819911	495680822274	1114433520849	1128229031556	589574114532	171960923160	29155354083	2901241395	165081321	4920750	59049			3612107166121
13	16777216	17779163471	782212447921	6676230149295	19317978800406	24779652162111	16355317614372	6067745422596	1332232683804	177404545494	14316489099	677311713	17124210	177147		75521580868855
14	67108864	124504475945	7875461969623	89137629284598	329116351254381	528767327481327	434155944002517	200759588408016	55505751414300	9496238961726	1018974265848	68010000678	2716903539	58989951	531441	1656028625052754

Technical details

Node id: n-b8827d5bb870ce1f · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-fb9a371d3330523a

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:     2,       1;
  2:     4,       5,        1;
  3:    16,      47,       25,         3;
  4:    64,     377,      391,       114,         9;
  5:   256,    2831,     5041,      2655,       486,       27;
  6:  1024,   20585,    58903,     49638,     15741,     1971,       81;
  7:  4096,  147167,   650785,    822003,    400770,    84672,     7695,     243;
  8: 16384, 1042457,  6949351,  12638394,   8878329,  2811078,   423306,   29160,    729;
  9: 65536, 7346351, 72620881, 185147175, 179968446, 80045469, 17745966, 1998918, 107892, 2187;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(3*n-6)*x)*P(n-1,x) + (3*n-6)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 4 + 5*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 4 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (3*n - 6)*x)*P[n - 1, x] + (3*n - 6)*x*D[P[n - 1, x], 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] = 2 + x; v[3] = 4 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (3*n - 6)*x)*v[i - 1] + (3*n - 6)*x*deriv(v[i - 1], x))/(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 (−∞, 0]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle T(n,k) read by rows, arising in enumeration of catafusenes.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.