A272867 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 - x^{2} + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(1 + 8\,x + 18\,x^{2} + 8\,x^{3} + x^{4} + 4\,n x + 18\,n x^{2} + 12\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	8	3																											12
2	1	16	54	32	5																									108
3	1	24	153	352	255	72	7																							864
4	1	32	300	1216	2270	1824	700	128	9																					6480
5	1	40	495	2880	8850	14544	12390	5760	1485	200	11																			46656
6	1	48	738	5600	24075	61344	92652	81792	43335	14000	2706	288	13																	326592
7	1	56	1029	9632	53305	185304	412629	587136	530523	308840	117271	28896	4459	392	15															2239488
8	1	64	1368	15232	103180	455616	1352008	2720512	3703158	3400640	2124584	911232	268268	53312	6840	512	17													15116544
9	1	72	1755	22656	181620	973728	3622668	9508608	17683758	23259760	21613482	14262912	6727812	2272032	544860	90624	9945	648	19											100776960
10	1	80	2190	32160	297825	1879488	8421000	27367680	65130210	113740000	145569908	136488000	94076970	47893440	18045000	5011968	1012605	144720	13870	800	21									665127936
11	1	88	2673	44000	462275	3357288	17608899	68369664	199111770	437417200	725629498	908190144	857562134	612384080	331852950	136739328	42764469	10071864	1756645	220000	18711	968	23							4353564672
12	1	96	3204	58432	686730	5642208	33923428	153343872	529415271	1407820480	2893624008	4599474048	5650614892	5366053056	3945850920	2252512768	1000006623	345023712	92077876	18807360	2884266	321376	24564	1152	25					28298170368
13	1	104	3783	75712	984230	9026160	61215154	315945344	1263078531	3952457080	9735097229	18915581568	29001406564	35073216416	33463161420	25220775424	15045150263	7114422744	2666499121	789863360	183645462	33095920	4527458	454272	31525	1352	27			182849716224
14	1	112	4410	96096	1369095	13864032	104715156	607820928	2763537777	9961071120	28669671030	66138954336	122474041599	182070530880	217240107480	208080606720	160158362091	99208431504	49520340870	19922142240	6448254813	1671507552	344064084	55456128	6845475	624624	39690	1568	29	1175462461440

Technical details

Node id: n-6b0327e8f50ac23f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2383397e4235b687

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,  8,    3;
  2: 1, 16,   54,    32,      5;
  3: 1, 24,  153,   352,    255,     72,       7;
  4: 1, 32,  300,  1216,   2270,   1824,     700,     128,        9;
  5: 1, 40,  495,  2880,   8850,  14544,   12390,    5760,     1485,      200,       11;
  6: 1, 48,  738,  5600,  24075,  61344,   92652,   81792,    43335,    14000,     2706,      288,      13;
  7: 1, 56, 1029,  9632,  53305, 185304,  412629,  587136,   530523,   308840,   117271,    28896,    4459,     392,     15;
  8: 1, 64, 1368, 15232, 103180, 455616, 1352008, 2720512,  3703158,  3400640,  2124584,   911232,  268268,   53312,   6840,   512,   17;
  9: 1, 72, 1755, 22656, 181620, 973728, 3622668, 9508608, 17683758, 23259760, 21613482, 14262912, 6727812, 2272032, 544860, 90624, 9945, 648, 19;

%F Formula

The row polynomials satisfy (1+4*n*x+(2*n-1)*x^2)*P(n,x) = (1+(4*n+8)*x+(18*n+18)*x^2+(12*n+8)*x^3+(2*n+1)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (4*n + 8)*x + (18*n + 18)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(1 + 4*n*x + (2*n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (4*n + 8)*x + (18*n + 18)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(1 + 4*n*x + (2*n - 1)*x^2)); 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 read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.