A160562 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 + 12\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(28\,x + 24\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(60\,x^{2} + 52\,x^{3}\right)\,P_{n-2}''(x) + \left(16\,x^{3} + 16\,x^{4}\right)\,P_{n-2}'''(x) + \left(2 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x - 4\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x^{2}\right)\,P_{n-1}''(x)\)

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

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

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	40	9													50
3	1	364	315	16												696
4	1	3280	8694	1344	25											13344
5	1	29524	224730	81312	4125	36										339728
6	1	265720	5684679	4383808	461175	10296	49									10805728
7	1	2391484	142714845	224912688	44204875	1909908	22295	64								416156160
8	1	21523360	3573251964	11274436992	3932020950	294753888	6367452	43520	81							19102398208
9	1	193710244	89379726660	558799860544	336110004750	41327330616	1477292180	18108672	78489	100						1027306112256
10	1	1743392200	2234929014549	27540090458496	28098035166850	5484605411376	305913467314	6003975680	45602109	133000	121					63671366621696
11	1	15690529804	55877147996175	1353437639603280	2318972239856250	704098425416760	59164533341550	1750454975104	20777791221	104311900	214291	144				4493337014036480
12	1	141214768240	1396964003596434	66417781492553920	189951497740236375	88535229500820960	10957162324872540	471128494579200	8220201240735	63308190000	220719730	331200	169			357738188501909504
13	1	1270932914164	34924417823139390	3256954784697091296	15489849100541261875	10986292926345276540	1972266717501798900	120315306969949440	2971910159524215	33002653183500	173940310830	437879520	494325	196		31863609612002824192
14	1	11438396227480	873113305177541619	159652872304065276864	1259766768994931417025	1351646826792557878440	348266640629773973835	29646965903343014400	1011156096486340395	15582982687545000	116640887428065	438641611200	822853395	716184	225	3150880055527771824128

Technical details

Node id: n-4a61d31ab117b16d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e07ab4f767b2303c

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,        40,           9;
  3: 1,       364,         315,           16;
  4: 1,      3280,        8694,         1344,           25;
  5: 1,     29524,      224730,        81312,         4125,          36;
  6: 1,    265720,     5684679,      4383808,       461175,       10296,         49;
  7: 1,   2391484,   142714845,    224912688,     44204875,     1909908,      22295,       64;
  8: 1,  21523360,  3573251964,  11274436992,   3932020950,   294753888,    6367452,    43520,    81;
  9: 1, 193710244, 89379726660, 558799860544, 336110004750, 41327330616, 1477292180, 18108672, 78489, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(1-12*x-9*x^2)*P(n-2,x) + (28*x+24*x^2+4*x^3)*d/dx P(n-2,x) + (60*x^2+52*x^3)*d^2/dx^2 P(n-2,x) + (16*x^3+16*x^4)*d^3/dx^3 P(n-2,x) + (2+4*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*x^2*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[-(1 - 12*x - 9*x^2)*P[n - 2, x] + (28*x + 24*x^2 + 4*x^3)*D[P[n - 2, x], x] + (60*x^2 + 52*x^3)*D[P[n - 2, x], {x, 2}] + (16*x^3 + 16*x^4)*D[P[n - 2, x], {x, 3}] + (2 + 4*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 - 12*x - 9*x^2)*v[i - 2] + (28*x + 24*x^2 + 4*x^3)*deriv(v[i - 2], x) + (60*x^2 + 52*x^3)*deriv(deriv(v[i - 2], x), x) + (16*x^3 + 16*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2 + 4*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*x^2*deriv(deriv(v[i - 1], x), x)); 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 of scaled central factorial numbers, T(n,k) = A008958(n,n-k).?
  - 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.