A286785: Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

A286785 ↗ nonnegative coefficients

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

Recurrence from A286785

\(\left(56 + 30\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(156 + 84\,x + 96\,n + 66\,n x + 15\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(2 + 26\,x - 28\,x^{2} + n + 7\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 14 + 14\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	14	14	2															30
1	42	72	27	2														143
2	132	330	220	44	2													728
3	429	1430	1430	520	65	2												3876
4	1430	6006	8190	4550	1050	90	2											21318
5	4862	24752	43316	33320	11900	1904	119	2										120175
6	16796	100776	217056	217056	108528	27132	3192	152	2									690690
7	58786	406980	1046520	1302336	854658	301644	55860	5040	189	2								4032015
8	208012	1634380	4903140	7354710	6056820	2826516	743820	106260	7590	230	2							23841480
9	742900	6537520	22472725	39657750	39657750	23377200	8182020	1669800	189750	11000	275	2						142498692
10	2674440	26075790	101235420	206220300	244208250	175829940	78146640	21312720	3474900	321750	15444	324	2					859515920
11	9694845	103791870	449764770	1041560520	1432145715	1227553470	669574620	232895520	50945895	6792786	522522	21112	377	2				5225264024
12	35357670	412506150	1975687350	5136787110	8072094030	8072094030	5264409150	2256175350	631729098	113387274	12598586	818090	28210	434	2			31983672534
13	129644790	1637618400	8597496600	24837212400	44029603800	50538327840	38605667100	19854343080	6872657220	1583822240	237573336	22342320	1241240	36960	495	2		196947587823
14	477638700	6495886320	37119350400	118107024000	233646504000	303740455200	267291600576	161549868480	67312445200	19232127200	3713790080	472664192	38118080	1832600	47600	560	2	1219199353190

Technical details

Node id: n-91d76a88748fc9ba · revision 0 · JSON
Definition source: oeis_catalog:A286785
Fingerprint: f6-ca8fcf8c70f1b4a8

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

Row sums ? A006013, A305574

Submission text ?

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

%e Example

Triangle begins:
  0:     14,      14,        2;
  1:     42,      72,       27,        2;
  2:    132,     330,      220,       44,        2;
  3:    429,    1430,     1430,      520,       65,        2;
  4:   1430,    6006,     8190,     4550,     1050,       90,       2;
  5:   4862,   24752,    43316,    33320,    11900,     1904,     119,       2;
  6:  16796,  100776,   217056,   217056,   108528,    27132,    3192,     152,      2;
  7:  58786,  406980,  1046520,  1302336,   854658,   301644,   55860,    5040,    189,     2;
  8: 208012, 1634380,  4903140,  7354710,  6056820,  2826516,  743820,  106260,   7590,   230,   2;
  9: 742900, 6537520, 22472725, 39657750, 39657750, 23377200, 8182020, 1669800, 189750, 11000, 275, 2;

%F Formula

The row polynomials satisfy (4*n^2+30*n+56)*P(n,x) = (15*n^2+96*n+156+(12*n^2+66*n+84)*x)*P(n-1,x) + (n+2+(7*n+26)*x-(8*n+28)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 14*x + 2*x^2.

%t Mathematica

P[0, x_] := 14 + 14*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n^2 + 96*n + 156 + (12*n^2 + 66*n + 84)*x)*P[n - 1, x] + (n + 2 + (7*n + 26)*x - (8*n + 28)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 30*n + 56)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 14 + 14*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^2 + 96*n + 156 + (12*n^2 + 66*n + 84)*x)*v[i - 1] + (n + 2 + (7*n + 26)*x - (8*n + 28)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 30*n + 56)); 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 Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)

Proof chains ? JSON

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

Used in 5

Related ?

Same fingerprint ?

A286785 → reverse

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.