A257623 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 5\,n\right)\,P_{n}(x) = \left(18 - 84\,x - 42\,x^{2} + 30\,n - 80\,n x + 65\,n x^{2} + 100\,n^{2} x + 150\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 30\,x^{3} + 50\,n x - 75\,n x^{2} + 25\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-12 + 8\,x + 5\,n - 40\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-10\,x + 10\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	9	96	27													132
3	27	1002	1503	108												2640
4	81	8988	39078	17976	405											66528
5	243	75630	750690	1000920	189075	1458										2018016
6	729	618648	12368205	36033120	20613675	1855944	5103									71495424
7	2187	4997298	186769881	1027887780	1284859725	373539762	17490543	17496								2895564672
8	6561	40144596	2675374704	25474722264	59103547350	38212083396	6242540976	160578384	59049							131909057280
9	19683	321721014	37068113124	576261487728	2250996868290	2701196241948	1008457603524	98848301664	1447744563	196830						6674598298368
10	59049	2575657680	502636476015	12250824510720	75543714339450	151266989549088	105761200075230	24501649021440	1507909428045	12878288400	649539					371350378054656
11	177147	20611520634	6719721541155	249332665817820	2319419594066550	7227006795216684	8431507927752798	3711071350506480	560998498090095	22399071803850	113363363487	2125764				22528589601982464
12	531441	164912714124	88994995925418	4918050163354680	66748281451238475	308121723755501832	556479523231684668	410828965007335776	120146906612229255	12295125408386700	326314985059866	989476284744	6908733			1479955040007155712
13	1594323	1319368674558	1171282353159222	94813882652447112	1830293483179765725	12071619588037956606	32023887353139508596	36598728403588009824	18107429382056934909	3660586966359531450	260738177294229558	4685129412636888	8575896384627	22320522		104653963543363153920
14	4782969	10555166224392	15351350930815617	1797229056388361184	48378169838859232845	443430253096217290728	1662381198653507940069	2781503358672688746624	2137347255411653065803	739050421827028817880	106431973645490312259	5391687169165083552	66522520700201007	73886163570744	71744535	7925793505684036190208

Technical details

Node id: n-214a184f3ede7fe7 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-ade7ffc8995f784b

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 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:     3,         6;
  2:     9,        96,          27;
  3:    27,      1002,        1503,          108;
  4:    81,      8988,       39078,        17976,           405;
  5:   243,     75630,      750690,      1000920,        189075,          1458;
  6:   729,    618648,    12368205,     36033120,      20613675,       1855944,          5103;
  7:  2187,   4997298,   186769881,   1027887780,    1284859725,     373539762,      17490543,       17496;
  8:  6561,  40144596,  2675374704,  25474722264,   59103547350,   38212083396,    6242540976,   160578384,      59049;
  9: 19683, 321721014, 37068113124, 576261487728, 2250996868290, 2701196241948, 1008457603524, 98848301664, 1447744563, 196830;

%F Formula

The row polynomials satisfy (5*n-2)*P(n,x) = (30*n+18+(100*n^2-80*n-84)*x-(125*n^3-150*n^2-65*n+42)*x^2)*P(n-2,x) + ((50*n+30)*x-(125*n^2+75*n)*x^2+(125*n^2+25*n-30)*x^3)*d/dx P(n-2,x) + (5*n-12+(50*n^2-40*n+8)*x)*P(n-1,x) + ((25*n-10)*x-(25*n-10)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 18 + (100*n^2 - 80*n - 84)*x - (125*n^3 - 150*n^2 - 65*n + 42)*x^2)*P[n - 2, x] + ((50*n + 30)*x - (125*n^2 + 75*n)*x^2 + (125*n^2 + 25*n - 30)*x^3)*D[P[n - 2, x], x] + (5*n - 12 + (50*n^2 - 40*n + 8)*x)*P[n - 1, x] + ((25*n - 10)*x - (25*n - 10)*x^2)*D[P[n - 1, x], x])/(5*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((30*n + 18 + (100*n^2 - 80*n - 84)*x - (125*n^3 - 150*n^2 - 65*n + 42)*x^2)*v[i - 2] + ((50*n + 30)*x - (125*n^2 + 75*n)*x^2 + (125*n^2 + 25*n - 30)*x^3)*deriv(v[i - 2], x) + (5*n - 12 + (50*n^2 - 40*n + 8)*x)*v[i - 1] + ((25*n - 10)*x - (25*n - 10)*x^2)*deriv(v[i - 1], x))/(5*n - 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.?
    
    ↑ Liu–Wang theorem (above)

References

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