A193726 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 15\,x + 7\,n + 15\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

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	6														9
2	21	87	90													198
3	147	924	1935	1350												4356
4	1029	8673	27405	38475	20250											95832
5	7203	76146	321930	680400	718875	303750										2108304
6	50421	641067	3395700	9591750	15238125	12909375	4556250									46382688
7	352947	5243784	33385905	118077750	250543125	318937500	225534375	68343750								1020419136
8	2470629	42000693	312358095	1327332825	3524968125	5990709375	6362803125	3861421875	1025156250							22449220992
9	17294403	331064286	2816517060	13976701200	44584769250	94809487500	134400262500	122472000000	65097421875	15377343750						493882861824
10	121060821	2576866047	24681583710	140084664300	521743902750	1332437951250	2362944150000	2873307937500	2292761953125	1084102734375	230660156250					10865422960128
11	847425747	19853974644	211424076675	1350816405750	5753477283750	17153224200000	36527178318750	55557317812500	59148952734375	41980148437500	17876162109375	3459902343750				239039305122816
12	5931980229	151689208713	1777778156385	12627075990375	60536587072500	206374728656250	512988611231250	936808899468750	1247402436328125	1181095330078125	754835361328125	292361748046875	51898535156250			5258864712701952
13	41523861603	1150804164426	14719785225390	115056204278400	613162249363125	2352671906681250	6686541208462500	14252491464750000	22783950546328125	26978703855468750	23000277480468750	13369062656250000	4748715966796875	778478027343750		115695023679442944
14	290667031221	8678487075027	120300559044120	1026190208329650	6017978809717875	25666137087215625	82095867059456250	200065558380187500	373275025795546875	530610185183203125	565682500195312500	438587600800781250	233776951611328125	76680085693359375	11677170410156250	2545290520947744768

Technical details

Node id: n-f929f0b56bd73253 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-4645d5c588e5d1ae

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:        1;
  1:        3,         6;
  2:       21,        87,         90;
  3:      147,       924,       1935,        1350;
  4:     1029,      8673,      27405,       38475,       20250;
  5:     7203,     76146,     321930,      680400,      718875,      303750;
  6:    50421,    641067,    3395700,     9591750,    15238125,    12909375,      4556250;
  7:   352947,   5243784,   33385905,   118077750,   250543125,   318937500,    225534375,     68343750;
  8:  2470629,  42000693,  312358095,  1327332825,  3524968125,  5990709375,   6362803125,   3861421875,  1025156250;
  9: 17294403, 331064286, 2816517060, 13976701200, 44584769250, 94809487500, 134400262500, 122472000000, 65097421875, 15377343750;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(15*n-15)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (15*n - 15)*x)*P[n - 1, 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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (15*n - 15)*x)*v[i - 1])/(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 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.