A257625 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-9 - 540\,x - 135\,x^{2} + 360\,n x + 144\,n x^{2} + 36\,n^{2} + 432\,n^{2} x + 1368\,n^{2} x^{2} - 288\,n^{3} x - 2880\,n^{3} x^{2} + 1296\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(144\,x - 540\,x^{2} - 180\,x^{3} + 72\,n x^{2} + 72\,n x^{3} - 144\,n^{2} x + 432\,n^{2} x^{2} + 2016\,n^{2} x^{3} + 288\,n^{3} x^{2} - 3168\,n^{3} x^{3} + 576\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(108\,x^{2} - 72\,x^{3} - 36\,x^{4} - 144\,n^{2} x^{2} - 288\,n^{2} x^{3} + 432\,n^{2} x^{4} + 576\,n^{3} x^{3} - 576\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 - 12\,x - 12\,n + 72\,n x - 144\,n^{2} x + 96\,n^{3} x\right)\,P_{n-1}(x) + \left(-12\,x - 12\,x^{2} - 24\,n x + 24\,n x^{2} + 48\,n^{2} x + 48\,n^{2} x^{2} - 96\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 12\,x\)

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	12														15
2	9	216	81													306
3	27	2484	5589	432												8532
4	81	24624	167670	98496	2025											292896
5	243	230364	3678534	6539616	1439775	8748										11897280
6	729	2105352	69172623	274663872	192146175	18948168	35721									557072640
7	2187	19061892	1193911731	9088227216	14200355025	4775646924	233508177	139968								29510853120
8	6561	171950688	19581356988	260895593088	766819171350	587015084448	106609610268	2751211008	531441							1743844515840
9	19683	1548894636	311130361980	6836448206016	34156546645050	49185427168872	20936622630924	2212482574080	31365116379	1968300						113671573585920
10	59049	13944539448	4844691089181	168455841284736	1338822821970450	3246238193145552	2624092731062082	673823365138944	43602219802629	348613486200	7144929					8100242428723200
11	177147	125515735380	74458753521921	3976824659065200	47992767533535150	182314150055663688	248149815353542242	122861484885849984	20132674836517575	827319483576900	3796850995245	25509168				626333417953689600
12	531441	1129690510992	1134673158318930	91058471530590528	1612707938424707175	9126298288009979424	19355685597576294876	16224530289795518976	5225173720496051247	569115447066190800	15255050239621170	40668858395712	89813529			52221001274936524800
13	1594323	10167374031228	17195481826615458	2039311295874121248	51653009458160312625	419605114502239815924	1313857508446086739884	1716058786541827578624	944111507630039585829	206612037832641250500	15422291675048041938	275127709225847328	429571552819383	312487308		4669652330312210841600
14	4782969	91506882841704	259647971767001595	44934849650754709056	1595375961057937251225	18086647794899712385752	80368031194427863410867	154445290788764482076160	132853276056095039515923	50240688319165867738200	7721619651520416295929	404413646856792381504	4875611914291474395	4483837259243496	1076168025	445765418097670147276800

Technical details

Node id: n-bfe4e5560512fd9d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5d3f47477444bdf9

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,         12;
  2:     9,        216,           81;
  3:    27,       2484,         5589,           432;
  4:    81,      24624,       167670,         98496,           2025;
  5:   243,     230364,      3678534,       6539616,        1439775,           8748;
  6:   729,    2105352,     69172623,     274663872,      192146175,       18948168,          35721;
  7:  2187,   19061892,   1193911731,    9088227216,    14200355025,     4775646924,      233508177,        139968;
  8:  6561,  171950688,  19581356988,  260895593088,   766819171350,   587015084448,   106609610268,    2751211008,      531441;
  9: 19683, 1548894636, 311130361980, 6836448206016, 34156546645050, 49185427168872, 20936622630924, 2212482574080, 31365116379, 1968300;

%F Formula

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (36*n^2-9-(288*n^3-432*n^2-360*n+540)*x+(1296*n^4-2880*n^3+1368*n^2+144*n-135)*x^2)*P(n-2,x) - ((144*n^2-144)*x-(288*n^3+432*n^2+72*n-540)*x^2-(576*n^4-3168*n^3+2016*n^2+72*n-180)*x^3)*d/dx P(n-2,x) - ((144*n^2-108)*x^2-(576*n^3-288*n^2-72)*x^3+(576*n^3-432*n^2+36)*x^4)*d^2/dx^2 P(n-2,x) - (12*n-6-(96*n^3-144*n^2+72*n-12)*x)*P(n-1,x) + ((48*n^2-24*n-12)*x-(96*n^3-48*n^2-24*n+12)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((36*n^2 - 9 - (288*n^3 - 432*n^2 - 360*n + 540)*x + (1296*n^4 - 2880*n^3 + 1368*n^2 + 144*n - 135)*x^2)*P[n - 2, x] - ((144*n^2 - 144)*x - (288*n^3 + 432*n^2 + 72*n - 540)*x^2 - (576*n^4 - 3168*n^3 + 2016*n^2 + 72*n - 180)*x^3)*D[P[n - 2, x], x] - ((144*n^2 - 108)*x^2 - (576*n^3 - 288*n^2 - 72)*x^3 + (576*n^3 - 432*n^2 + 36)*x^4)*D[P[n - 2, x], {x, 2}] - (12*n - 6 - (96*n^3 - 144*n^2 + 72*n - 12)*x)*P[n - 1, x] + ((48*n^2 - 24*n - 12)*x - (96*n^3 - 48*n^2 - 24*n + 12)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 4*n + 1)]];
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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((36*n^2 - 9 - (288*n^3 - 432*n^2 - 360*n + 540)*x + (1296*n^4 - 2880*n^3 + 1368*n^2 + 144*n - 135)*x^2)*v[i - 2] - ((144*n^2 - 144)*x - (288*n^3 + 432*n^2 + 72*n - 540)*x^2 - (576*n^4 - 3168*n^3 + 2016*n^2 + 72*n - 180)*x^3)*deriv(v[i - 2], x) - ((144*n^2 - 108)*x^2 - (576*n^3 - 288*n^2 - 72)*x^3 + (576*n^3 - 432*n^2 + 36)*x^4)*deriv(deriv(v[i - 2], x), x) - (12*n - 6 - (96*n^3 - 144*n^2 + 72*n - 12)*x)*v[i - 1] + ((48*n^2 - 24*n - 12)*x - (96*n^3 - 48*n^2 - 24*n + 12)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 4*n + 1)); 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(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - 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(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.