A225473 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 16\,x - 32\,x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 12\,x^{2} - 16\,x^{3}\right)\,P_{n-2}'(x) + \left(2 + 16\,x\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 16\,x\)

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	16														19
3	9	160	288													457
4	27	1264	6048	6144												13483
5	81	9280	89280	221184	153600											473425
6	243	66256	1149120	5222400	8448000	4423680										19309699
7	729	467680	13832928	102850560	291072000	345047040	144506880									897777817
8	2187	3285424	160580448	1837860864	8101632000	16318955520	15173222400	5284823040								46880361883
9	6561	23032960	1825522560	30993629184	199877529600	608662978560	942762885120	718735933440	214035333120							2716916851105
10	19683	161335696	20495341440	503848919040	4572513792000	19755721359360	45337588531200	56758999449600	36600041963520	9512681472000						173062052183539
11	59049	1129664800	228352798368	7994967736320	99473985024000	586069988474880	1869468454748160	3417589363507200	3579740946432000	1997663109120000	460413783244800					12018644080809577
12	177147	7908598384	2532214748448	124796042409984	2089879908864000	16344460503613440	69620601235046400	174314402328084480	263703302347161600	237008458874880000	116484687160934400	24108939922636800				903802068447155083
13	531441	55363023040	27996717003840	1925961217449984	42827619328665600	436111132958392320	2413675276464291840	7949871317338030080	16289348876438077440	20963476667105280000	16480050957464371200	7232681976791040000	1358136948975206400			73168134786161363185
14	1594323	387549664336	308960421456960	29486681557831680	861873797680896000	11263991494708592640	79415529474510028800	334717563805315891200	892096999027522928640	1541546651192131584000	1723274461858863513600	1202891448500040499200	476706069090297446400	81906105230504755200		6344710489501106682979
15	4782969	2712873159520	3405540529984608	448891379025223680	17112769194882816000	283893769751507435520	2512176351305584803840	13280583841604016537600	44779456300078596096000	99769074908827484160000	148711659777630299750400	146772815355020574720000	92074894455774117888000	33253878723584930611200	5265392479103877120000	586721391031508305089337

Technical details

Node id: n-87220c9644b39788 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-661188e67d3889fb

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,        16;
   3:     9,       160,         288;
   4:    27,      1264,        6048,         6144;
   5:    81,      9280,       89280,       221184,        153600;
   6:   243,     66256,     1149120,      5222400,       8448000,        4423680;
   7:   729,    467680,    13832928,    102850560,     291072000,      345047040,      144506880;
   8:  2187,   3285424,   160580448,   1837860864,    8101632000,    16318955520,    15173222400,     5284823040;
   9:  6561,  23032960,  1825522560,  30993629184,  199877529600,   608662978560,   942762885120,   718735933440,   214035333120;
  10: 19683, 161335696, 20495341440, 503848919040, 4572513792000, 19755721359360, 45337588531200, 56758999449600, 36600041963520, 9512681472000;

%F Formula

The row polynomials satisfy P(n,x) = (3+16*x-32*x^2)*P(n-2,x) + (4*x-12*x^2-16*x^3)*d/dx P(n-2,x) + (2+16*x)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 16*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 16*x;
P[n_, x_] := P[n, x] = Expand[(3 + 16*x - 32*x^2)*P[n - 2, x] + (4*x - 12*x^2 - 16*x^3)*D[P[n - 2, x], x] + (2 + 16*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; v[2] = 3 + 16*x; for(n = 3, N, my(i = n); v[i] = (3 + 16*x - 32*x^2)*v[i - 2] + (4*x - 12*x^2 - 16*x^3)*deriv(v[i - 2], x) + (2 + 16*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.