A193729 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 4\,x + 7\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x + 4\,n x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	21	34	8													63
3	147	458	256	32												893
4	1029	5626	5672	1632	128											14087
5	7203	66002	107584	53696	9472	512										244469
6	50421	754834	1877768	1450560	432640	51712	2048									4619983
7	352947	8504858	31185856	35071712	15752960	3126784	270336	8192								94273645
8	2470629	94965226	501807272	791105952	502604928	147435008	20887552	1368064	32768							2062677399
9	17294403	1054500002	7906969984	17038242176	14724337152	5991164928	1237254144	131432448	6750208	131072						48108076517
10	121060821	11668677634	122822549768	355354481280	406812723200	220658801664	62319538176	9549152256	788987904	32636928	524288					1190129133919
11	847425747	128839697258	1889012957056	7243025343392	10778110558720	7585038537728	2814540890112	583498481664	68967137280	4559339520	155189248	2097152				31096597654877
12	5931980229	1420626372826	28850553144872	145173533352672	276868644224128	247908482753536	117590921744384	31680610418688	5023712280576	471920148480	25531252736	727711744	8388608			855021203773479
13	41523861603	15650618022002	438440802664384	2873699346280256	6948672950565632	7801003611241984	4636952505090048	1579185051631616	322647220617216	40387415506944	3087649472512	139238309888	3372220416	33554432		24659911339038933
14	290667031221	172322893688434	6639214512053768	56354050789982400	171314275248130560	238421789305796096	174949542102759424	73819286827466752	18899981810597888	3027247749267456	306669187235840	19451751694336	742425362432	15468593152	134217728	743924880873877487

Technical details

Node id: n-ded199f7951404aa · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-eafa5bb1fc5dff02

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:        3,          2;
  2:       21,         34,          8;
  3:      147,        458,        256,          32;
  4:     1029,       5626,       5672,        1632,         128;
  5:     7203,      66002,     107584,       53696,        9472,        512;
  6:    50421,     754834,    1877768,     1450560,      432640,      51712,       2048;
  7:   352947,    8504858,   31185856,    35071712,    15752960,    3126784,     270336,      8192;
  8:  2470629,   94965226,  501807272,   791105952,   502604928,  147435008,   20887552,   1368064,   32768;
  9: 17294403, 1054500002, 7906969984, 17038242176, 14724337152, 5991164928, 1237254144, 131432448, 6750208, 131072;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(4*n-4)*x)*P(n-1,x) + (4*n-4)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (4*n - 4)*x)*P[n - 1, x] + (4*n - 4)*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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (4*n - 4)*x)*v[i - 1] + (4*n - 4)*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 Mirror of the triangle A193728.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
  - 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.