A286785 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(42 + 84\,x + 26\,n + 94\,n x + 4\,n^{2} + 34\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(116 + 360\,x + 196\,x^{2} + 80\,n + 390\,n x + 238\,n x^{2} + 14\,n^{2} + 135\,n^{2} x + 94\,n^{2} x^{2} + 15\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2 + 22\,x + 46\,x^{2} - 70\,x^{3} + n + 10\,n x + 37\,n x^{2} - 48\,n x^{3} + n^{2} x + 7\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 14 + 28\,x + 6\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	14	28	6															48
1	42	144	81	8														275
2	132	660	660	176	10													1638
3	429	2860	4290	2080	325	12												9996
4	1430	12012	24570	18200	5250	540	14											62016
5	4862	49504	129948	133280	59500	11424	833	16										389367
6	16796	201552	651168	868224	542640	162792	22344	1216	18									2466750
7	58786	813960	3139560	5209344	4273290	1809864	391020	40320	1701	20								15737865
8	208012	3268760	14709420	29418840	30284100	16959096	5206740	850080	68310	2300	22							100975680
9	742900	13075040	67418175	158631000	198288750	140263200	57274140	13358400	1707750	110000	3025	24						650872404
10	2674440	52151580	303706260	824881200	1221041250	1054979640	547026480	170501760	31274100	3217500	169884	3888	26					4211628008
11	9694845	207583740	1349294310	4166242080	7160728575	7365320820	4687022340	1863164160	458513055	67927860	5747742	253344	4901	28				27341497800
12	35357670	825012300	5927062050	20547148440	40360470150	48432564180	36850864050	18049402800	5685561882	1133872740	138584446	9817080	366730	6076	30			177996090624
13	129644790	3275236800	25792489800	99348849600	220148019000	303229967040	270239669700	158834744640	61853914980	15838222400	2613306696	268107840	16136120	517440	7425	32		1161588834303
14	477638700	12991772640	111358051200	472428096000	1168232520000	1822442731200	1871041204032	1292398947840	605812006800	192321272000	40851690880	5671970304	495535040	25656400	714000	8960	34	7596549816030

Technical details

Node id: n-ed37a889ab35bb0d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9a1a7505fd4999af

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:     14,       28,        6;
  1:     42,      144,       81,         8;
  2:    132,      660,      660,       176,        10;
  3:    429,     2860,     4290,      2080,       325,        12;
  4:   1430,    12012,    24570,     18200,      5250,       540,       14;
  5:   4862,    49504,   129948,    133280,     59500,     11424,      833,       16;
  6:  16796,   201552,   651168,    868224,    542640,    162792,    22344,     1216,      18;
  7:  58786,   813960,  3139560,   5209344,   4273290,   1809864,   391020,    40320,    1701,     20;
  8: 208012,  3268760, 14709420,  29418840,  30284100,  16959096,  5206740,   850080,   68310,   2300,   22;
  9: 742900, 13075040, 67418175, 158631000, 198288750, 140263200, 57274140, 13358400, 1707750, 110000, 3025, 24;

%F Formula

The row polynomials satisfy (4*n^2+26*n+42+(4*n^3+34*n^2+94*n+84)*x)*P(n,x) = (14*n^2+80*n+116+(15*n^3+135*n^2+390*n+360)*x+(12*n^3+94*n^2+238*n+196)*x^2)*P(n-1,x) + (n+2+(n^2+10*n+22)*x+(7*n^2+37*n+46)*x^2-(8*n^2+48*n+70)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 28*x + 6*x^2.

%t Mathematica

P[0, x_] := 14 + 28*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^2 + 80*n + 116 + (15*n^3 + 135*n^2 + 390*n + 360)*x + (12*n^3 + 94*n^2 + 238*n + 196)*x^2)*P[n - 1, x] + (n + 2 + (n^2 + 10*n + 22)*x + (7*n^2 + 37*n + 46)*x^2 - (8*n^2 + 48*n + 70)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 26*n + 42 + (4*n^3 + 34*n^2 + 94*n + 84)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 14 + 28*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^2 + 80*n + 116 + (15*n^3 + 135*n^2 + 390*n + 360)*x + (12*n^3 + 94*n^2 + 238*n + 196)*x^2)*v[i - 1] + (n + 2 + (n^2 + 10*n + 22)*x + (7*n^2 + 37*n + 46)*x^2 - (8*n^2 + 48*n + 70)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 26*n + 42 + (4*n^3 + 34*n^2 + 94*n + 84)*x)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.?
  - 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

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.