A166962 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n\right)\,P_{n}(x) = \left(-40\,x + 30\,x^{2} - 96\,n x + 207\,n x^{2} + 72\,n^{2} x - 540\,n^{2} x^{2} + 243\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(16\,x - 44\,x^{2} + 34\,x^{3} + 48\,n x - 183\,n x^{2} + 297\,n x^{3} + 171\,n^{2} x^{2} - 594\,n^{2} x^{3} + 108\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(26\,x^{2} - 32\,x^{3} + 6\,x^{4} + 78\,n x^{2} - 237\,n x^{3} + 63\,n x^{4} + 387\,n^{2} x^{3} - 108\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(6\,x^{3} - 6\,x^{4} + 18\,n x^{3} - 63\,n x^{4} + 108\,n^{2} x^{4}\right)\,P_{n-2}'''(x) + \left(-2 + 16\,x + 3\,n - 48\,n x + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(-16\,x + 8\,x^{2} + 15\,n x + 12\,n x^{2} - 36\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(-6\,x^{2} + 9\,n x^{2}\right)\,P_{n-1}''(x)\)

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

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

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	48	9													58
3	1	412	621	16												1050
4	1	3336	19530	5056	25											27948
5	1	26740	485190	445280	33225	36										990472
6	1	213984	10971135	26436800	7029875	194832	49									44846676
7	1	1711948	238097259	1311027120	870016875	89565012	1067269	64								2511485548
8	1	13695672	5073228216	59213629056	83181335250	21126484152	995302308	5587200	81							169609261936
9	1	109565476	107215728300	2539907537536	6887127517650	3585297074472	419930130144	10083254016	28303425	100						13549699111120
10	1	876523920	2257693352325	105789627763840	523066793667850	502867937155968	119290547210226	7259454406656	95699599173	139798000	121					1260628769478080
11	1	7012191484	47466781405785	4331927037323600	37635860042792250	62586429441725304	26816988965912250	3303168777181824	113409879648165	866177013100	676663581	144				134836124791857488
12	1	56097532008	997291509877494	175639872389587520	2615546802676949375	7200601272561287664	5185214138306503084	1141778045622657024	79864671707906823	1642651534633000	7567261582078	3221201088	169			16401292372889717328
13	1	448780256212	20947413168625986	7080556738794527136	177694786591046163375	785417407062253793244	905251485167572357444	329630903216949018880	41090556558137839911	1741721111429077900	22456519768547898	64353916777248	15121830841	196		2247950885196938816272
14	1	3590242049856	439933037497475355	284488424303306922816	11893125595453360527525	82564724873327345458704	147124456517599926226827	84027163274671637104128	17171968351376084918355	1299347225928790501600	35054946323854228585	293507542398083904	535958903047607	70151088336	225	344401063189183497633824

Technical details

Node id: n-aaf5f334c8903a15 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e468febdff774e05

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,        48,            9;
  3: 1,       412,          621,            16;
  4: 1,      3336,        19530,          5056,            25;
  5: 1,     26740,       485190,        445280,         33225,            36;
  6: 1,    213984,     10971135,      26436800,       7029875,        194832,           49;
  7: 1,   1711948,    238097259,    1311027120,     870016875,      89565012,      1067269,          64;
  8: 1,  13695672,   5073228216,   59213629056,   83181335250,   21126484152,    995302308,     5587200,       81;
  9: 1, 109565476, 107215728300, 2539907537536, 6887127517650, 3585297074472, 419930130144, 10083254016, 28303425, 100;

%F Formula

The row polynomials satisfy (3*n-2)*P(n,x) = ((72*n^2-96*n-40)*x+(243*n^3-540*n^2+207*n+30)*x^2)*P(n-2,x) + ((48*n+16)*x+(171*n^2-183*n-44)*x^2+(108*n^3-594*n^2+297*n+34)*x^3)*d/dx P(n-2,x) + ((78*n+26)*x^2+(387*n^2-237*n-32)*x^3-(108*n^2-63*n-6)*x^4)*d^2/dx^2 P(n-2,x) + ((18*n+6)*x^3+(108*n^2-63*n-6)*x^4)*d^3/dx^3 P(n-2,x) + (3*n-2+(36*n^2-48*n+16)*x)*P(n-1,x) + ((15*n-16)*x-(36*n^2-12*n-8)*x^2)*d/dx P(n-1,x) + (9*n-6)*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[Together[(((72*n^2 - 96*n - 40)*x + (243*n^3 - 540*n^2 + 207*n + 30)*x^2)*P[n - 2, x] + ((48*n + 16)*x + (171*n^2 - 183*n - 44)*x^2 + (108*n^3 - 594*n^2 + 297*n + 34)*x^3)*D[P[n - 2, x], x] + ((78*n + 26)*x^2 + (387*n^2 - 237*n - 32)*x^3 - (108*n^2 - 63*n - 6)*x^4)*D[P[n - 2, x], {x, 2}] + ((18*n + 6)*x^3 + (108*n^2 - 63*n - 6)*x^4)*D[P[n - 2, x], {x, 3}] + (3*n - 2 + (36*n^2 - 48*n + 16)*x)*P[n - 1, x] + ((15*n - 16)*x - (36*n^2 - 12*n - 8)*x^2)*D[P[n - 1, x], x] + (9*n - 6)*x^2*D[P[n - 1, x], {x, 2}])/(3*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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((72*n^2 - 96*n - 40)*x + (243*n^3 - 540*n^2 + 207*n + 30)*x^2)*v[i - 2] + ((48*n + 16)*x + (171*n^2 - 183*n - 44)*x^2 + (108*n^3 - 594*n^2 + 297*n + 34)*x^3)*deriv(v[i - 2], x) + ((78*n + 26)*x^2 + (387*n^2 - 237*n - 32)*x^3 - (108*n^2 - 63*n - 6)*x^4)*deriv(deriv(v[i - 2], x), x) + ((18*n + 6)*x^3 + (108*n^2 - 63*n - 6)*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (3*n - 2 + (36*n^2 - 48*n + 16)*x)*v[i - 1] + ((15*n - 16)*x - (36*n^2 - 12*n - 8)*x^2)*deriv(v[i - 1], x) + (9*n - 6)*x^2*deriv(deriv(v[i - 1], x), x))/(3*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.?
  - 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.