A225472 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 6\,x\right)\,P_{n-1}(x) + \left(3\,x + 3\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	6														8
3	4	42	54													100
4	8	234	810	648												1700
5	16	1218	8586	16848	9720											36388
6	32	6186	79650	288360	388800	174960										937988
7	64	31122	692874	4127760	9768600	9972720	3674160									28267300
8	128	155994	5823090	53719848	198676800	345371040	282910320	88179840								974837060
9	256	780738	47988666	660795408	3587272920	9447490080	12910998240	8817984000	2380855680							37854165988
10	512	3905226	390935970	7844613480	60133752000	225178243920	456617256480	512677589760	299987815680	71425670400						1634259783428
11	1024	19529202	3162634794	90981979920	959541730200	4910437682640	13861088251920	22750398720000	21641978131200	11070978912000	2357047123200					77645634696100
12	2048	97652154	25476841170	1038753396648	14798313921600	100749191748480	380340956373840	855925288606080	1176952196851200	970317732384000	440767812038400	84853696435200				4025769516250820
13	4096	488273058	204693598746	11732009457168	222757695852120	1979105910312960	9722552154194880	28814464590912000	53710739910495360	63447780144672000	46125055153900800	18837520608614400	3309294160972800			226181207521260388
14	8192	2441389866	1641943247490	131508427213800	3294587883787200	37654439000658480	236012267200469760	896073937291653120	2174469781627503360	3451307821510348800	3569778509699001600	2319815206841932800	860416481852928000	138990354760857600		13687946540481000068
15	16384	12206998482	13157518488714	1466296018321680	48096856781227800	699428044919363760	5510988563023223280	26273994970519296000	80730210629189721600	165322020272625216000	228126070420209561600	209705558588631705600	123168619377246643200	41836096783018137600	6254565964238592000	887677129936146513700

Technical details

Node id: n-8256b7d953acaeb6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-fe60192346c5112b

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:   2,       6;
   3:   4,      42,        54;
   4:   8,     234,       810,        648;
   5:  16,    1218,      8586,      16848,        9720;
   6:  32,    6186,     79650,     288360,      388800,       174960;
   7:  64,   31122,    692874,    4127760,     9768600,      9972720,      3674160;
   8: 128,  155994,   5823090,   53719848,   198676800,    345371040,    282910320,     88179840;
   9: 256,  780738,  47988666,  660795408,  3587272920,   9447490080,  12910998240,   8817984000,   2380855680;
  10: 512, 3905226, 390935970, 7844613480, 60133752000, 225178243920, 456617256480, 512677589760, 299987815680, 71425670400;

%F Formula

The row polynomials satisfy P(n,x) = (2+6*x)*P(n-1,x) + (3*x+3*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[(2 + 6*x)*P[n - 1, x] + (3*x + 3*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] = (2 + 6*x)*v[i - 1] + (3*x + 3*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_3(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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[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.