A193726 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 5\,x + 7\,n + 5\,n x\right)\,P_{n-1}(x) + \left(-5\,x + 5\,n x\right)\,P_{n-1}'(x)\)

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

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

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	39	10													70
3	147	573	365	50												1135
4	1029	7611	9070	2925	250											20885
5	7203	96477	192245	109700	21375	1250										428250
6	50421	1193739	3750550	3374625	1125625	146875	6250									9648085
7	352947	14576973	69728045	92994500	47265000	10328125	965625	31250								236242465
8	2470629	176688411	1258261630	2394519225	1741127500	566825000	87368750	6140625	156250							6233558020
9	17294403	2132614077	22273889765	58970731100	58983038625	26844037500	6066768750	694750000	38046875	781250						176021952345
10	121060821	25677840939	389319196390	1408725533025	1887395698375	1153924393125	358690631250	59513343750	5261953125	230859375	3906250					5288864416425
11	847425747	308739395373	6746815543325	32938557708500	58003311521250	46362559071875	19041175321875	4293013593750	544878515625	38314453125	1376953125	19531250				168279609034820
12	5931980229	3709109873211	116239561213390	758382347303625	1730782199616250	1773618447906250	936336282268750	275512447546875	47074358203125	4716744140625	270058593750	8095703125	97656250			5646655682005455
13	41523861603	44538978379677	1994618089993685	17265609446746700	50523031126156875	65409701331081250	43512534683475000	16253204208312500	3590057073281250	480642486328125	38977060546875	1852226562500	47021484375	488281250		199114855744491665
14	290667031221	534675359864139	34131202421791030	389816498278395825	1450449887639969125	2345725598225384375	1937012289943981250	900197250166500000	249998703485781250	42943694655468750	4624904882812500	309723349609375	12411572265625	270263671875	2441406250	7355757403353932590

Technical details

Node id: n-3f7299f6bf79f102 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-de53a882f20db64c

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,         39,          10;
  3:      147,        573,         365,          50;
  4:     1029,       7611,        9070,        2925,         250;
  5:     7203,      96477,      192245,      109700,       21375,        1250;
  6:    50421,    1193739,     3750550,     3374625,     1125625,      146875,       6250;
  7:   352947,   14576973,    69728045,    92994500,    47265000,    10328125,     965625,     31250;
  8:  2470629,  176688411,  1258261630,  2394519225,  1741127500,   566825000,   87368750,   6140625,   156250;
  9: 17294403, 2132614077, 22273889765, 58970731100, 58983038625, 26844037500, 6066768750, 694750000, 38046875, 781250;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(5*n-5)*x)*P(n-1,x) + (5*n-5)*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 + (5*n - 5)*x)*P[n - 1, x] + (5*n - 5)*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 + (5*n - 5)*x)*v[i - 1] + (5*n - 5)*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 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.?
  - 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
  - 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.