A380114 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

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	8	16	4													28
3	32	112	64	8												216
4	128	736	736	208	16											1824
5	512	4672	7360	3552	608	32										16736
6	2048	29056	68224	50240	14400	1664	64									165696
7	8192	178432	603904	638848	273280	52096	4352	128								1759232
8	32768	1086976	5188096	7596288	4557056	1275904	173824	11008	256							19922176
9	131072	6587392	43678720	86339072	69877248	26976768	5332992	545792	27136	512						239496704
10	524288	39786496	362604544	950748160	1011205120	517429248	139281408	20490240	1634304	65536	1024					3043770368
11	2097152	239767552	2980409344	10232690688	14035957760	9266419712	3263361024	647387136	73666560	4710400	155648	2048				40746625024
12	8388608	1442799616	24322809856	108287725568	188896874496	157801791488	70746615808	18179690496	2768105472	250961920	13156352	364544	4096			572719288320
13	33554432	8673574912	197468078080	1131522875392	2483337945088	2587018829824	1447549435904	468727660544	91721490432	11057373184	817676288	35790848	843776	8192		8427965136896
14	134217728	52108558336	1597091774464	11710164910080	32063101091840	41184939507712	28334828634112	11332196761600	2771885129728	426705190912	41738977280	2565914624	95207424	1933312	16384	129517557825536

Technical details

Node id: n-bd5fa5857f8c49a3 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-a546e680aefa4794

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,       2;
  2:      8,      16,        4;
  3:     32,     112,       64,        8;
  4:    128,     736,      736,      208,       16;
  5:    512,    4672,     7360,     3552,      608,       32;
  6:   2048,   29056,    68224,    50240,    14400,     1664,      64;
  7:   8192,  178432,   603904,   638848,   273280,    52096,    4352,    128;
  8:  32768, 1086976,  5188096,  7596288,  4557056,  1275904,  173824,  11008,   256;
  9: 131072, 6587392, 43678720, 86339072, 69877248, 26976768, 5332992, 545792, 27136, 512;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (2*n - 2)*x)*P[n - 1, x] + (2*n - 2)*x*D[P[n - 1, x], x])/(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; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (2*n - 2)*x)*v[i - 1] + (2*n - 2)*x*deriv(v[i - 1], 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)

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 read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is defined in A357…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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 ?

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
[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.