A191935 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-2 - n + 4\,n x - 2\,n^{2} x + 2\,n^{2} x^{2} - 2\,n^{3} x + n^{3} x^{2} - 2\,n^{4} x^{2} - n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x^{2} + n x^{2} - 4\,n x^{3} + 2\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{3} x^{3} + 2\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{3} - n x^{3} - 2\,n x^{4} - 3\,n^{2} x^{4} - n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(2 + 2\,n + 2\,n^{2} x + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(n x^{3}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	16	12													29
3	1	40	156	32												229
4	1	80	876	1280	80											2317
5	1	140	3276	15296	9680	192										28585
6	1	224	9576	102656	239120	69888	448									421913
7	1	336	23688	485696	2805520	3513216	489664	1024								7319145
8	1	480	51912	1812224	21019120	70845696	49674688	3358720	2304							146765145
9	1	660	103752	5688320	116160880	827534016	1702740928	684611584	22673664	5120						3359518925
10	1	880	192852	15648512	514343280	6682042368	30666431488	39604404224	9264930048	151162880	11264					86899167797
11	1	1144	338052	38790752	1922709360	41245910784	358086507520	1091024912384	900364025088	123683563520	997686272	24576				2517364469453
12	1	1456	564564	88371712	6286668960	207367999488	3052819345408	18279177273344	37722454818048	20131773009920	1633620724736	6530334720	53248			81040119165605
13	1	1820	905268	187734976	18437779360	886328247168	20471731302400	213659615379456	901413580983552	1277546933611520	444532626942976	21392110731264	42447228928	114688		2879964000963377
14	1	2240	1402128	376030720	49414050400	3320115122688	113536197255040	1898190648262656	14361969349889280	43343514046177280	42603581436123136	9720285789487104	278139886735360	274274516992	245760	112322861535300785

Technical details

Node id: n-aba11d16698c5dd3 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4ee3605589a7d639

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: 1,   4;
  2: 1,  16,     12;
  3: 1,  40,    156,      32;
  4: 1,  80,    876,    1280,        80;
  5: 1, 140,   3276,   15296,      9680,       192;
  6: 1, 224,   9576,  102656,    239120,     69888,        448;
  7: 1, 336,  23688,  485696,   2805520,   3513216,     489664,      1024;
  8: 1, 480,  51912, 1812224,  21019120,  70845696,   49674688,   3358720,     2304;
  9: 1, 660, 103752, 5688320, 116160880, 827534016, 1702740928, 684611584, 22673664, 5120;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n+2+(2*n^3+2*n^2-4*n)*x+(n^5+2*n^4-n^3-2*n^2)*x^2)*P(n-2,x) + ((2*n^2+n-6)*x^2+(2*n^4+4*n^3-2*n^2-4*n)*x^3)*d/dx P(n-2,x) - ((n+2)*x^3+(n^3+3*n^2+2*n)*x^4)*d^2/dx^2 P(n-2,x) + (2*n+2+(2*n^3+2*n^2)*x)*P(n-1,x) - (2*n^2-n)*x^2*d/dx P(n-1,x) + n*x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 2 + (2*n^3 + 2*n^2 - 4*n)*x + (n^5 + 2*n^4 - n^3 - 2*n^2)*x^2)*P[n - 2, x] + ((2*n^2 + n - 6)*x^2 + (2*n^4 + 4*n^3 - 2*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] - ((n + 2)*x^3 + (n^3 + 3*n^2 + 2*n)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n + 2 + (2*n^3 + 2*n^2)*x)*P[n - 1, x] - (2*n^2 - n)*x^2*D[P[n - 1, x], x] + n*x^3*D[P[n - 1, x], {x, 2}])/(n)]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 2 + (2*n^3 + 2*n^2 - 4*n)*x + (n^5 + 2*n^4 - n^3 - 2*n^2)*x^2)*v[i - 2] + ((2*n^2 + n - 6)*x^2 + (2*n^4 + 4*n^3 - 2*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) - ((n + 2)*x^3 + (n^3 + 3*n^2 + 2*n)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n + 2 + (2*n^3 + 2*n^2)*x)*v[i - 1] - (2*n^2 - n)*x^2*deriv(v[i - 1], x) + n*x^3*deriv(deriv(v[i - 1], x), x))/(n)); 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 of Legendre-Stirling numbers of the second kind.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.