A225473 → θ+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	8														11
3	9	80	96													185
4	27	632	2016	1536												4211
5	81	4640	29760	55296	30720											120497
6	243	33128	383040	1305600	1689600	737280										4148891
7	729	233840	4610976	25712640	58214400	57507840	20643840									166924265
8	2187	1642712	53526816	459465216	1620326400	2719825920	2167603200	660602880								7682995331
9	6561	11516480	608507520	7748407296	39975505920	101443829760	134680412160	89841991680	23781703680							398091881057
10	19683	80667848	6831780480	125962229760	914502758400	3292620226560	6476798361600	7094874931200	4066671329280	951268147200						22929610452011
11	59049	564832400	76117599456	1998741934080	19894797004800	97678331412480	267066922106880	427198670438400	397748994048000	199766310912000	41855798476800					1453285248824345
12	177147	3954299192	844071582816	31199010602496	417975981772800	2724076750602240	9945800176435200	21789300291010560	29300366927462400	23700845887488000	10589517014630400	2009078326886400				100509008392949651
13	531441	27681511520	9332239001280	481490304362496	8565523865733120	72685188826398720	344810753780613120	993733914667253760	1809927652937564160	2096347666710528000	1498186450678579200	602723498065920000	104472072998092800			7531943572756089617
14	1594323	193774832168	102986807152320	7371670389457920	172374759536179200	1877331915784765440	11345075639215718400	41839695475664486400	99121888780835880960	154154665119213158400	156661314714442137600	100240954041670041600	36669697622330572800	5850436087893196800		607940909007559174331
15	4782969	1356436579760	1135180176661536	112222844756305920	3422553838976563200	47315628291917905920	358882335900797829120	1660072980200502067200	4975495144453177344000	9976907490882748416000	13519241797966390886400	12231067946251714560000	7082684188905701376000	2375277051684637900800	351026165273591808000	52581506643031530986825

Technical details

Node id: n-d457f4ac0217023f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-084952726d09af56

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     3,        8;
   3:     9,       80,         96;
   4:    27,      632,       2016,         1536;
   5:    81,     4640,      29760,        55296,        30720;
   6:   243,    33128,     383040,      1305600,      1689600,        737280;
   7:   729,   233840,    4610976,     25712640,     58214400,      57507840,      20643840;
   8:  2187,  1642712,   53526816,    459465216,   1620326400,    2719825920,    2167603200,     660602880;
   9:  6561, 11516480,  608507520,   7748407296,  39975505920,  101443829760,  134680412160,   89841991680,   23781703680;
  10: 19683, 80667848, 6831780480, 125962229760, 914502758400, 3292620226560, 6476798361600, 7094874931200, 4066671329280, 951268147200;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + 8*x)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3 + 8*x)*v[i - 1] + (4*x + 4*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? 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
[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.