A132166 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(6 + 5\,n + n^{2}\right)\,P_{n}(x) = \left(-252 - 336\,x + 22\,x^{2} - 288\,n - 48\,n x - 36\,n^{2}\right)\,P_{n-2}(x) + \left(-3024\,x + 24\,x^{2} + 26\,x^{3} - 432\,n x + 24\,n x^{2}\right)\,P_{n-2}'(x) + \left(-1296\,x^{2} + 144\,x^{3} + 5\,x^{4}\right)\,P_{n-2}''(x) + \left(108 + 16\,x + 78\,n + 8\,n x + 12\,n^{2}\right)\,P_{n-1}(x) + \left(144\,x - 8\,x^{2} + 72\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 336 + 168\,x + 9\,x^{2}\)

\(P_{1}(x) = 4536 + 4452\,x + 567\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	336	168	9								513
1	4536	4452	567	16							9571
2	54432	92736	20979	1344	25						169516
3	598752	1649088	586845	63840	2625	36					2901186
4	6158592	26127360	13662432	2237760	152250	4536	49				48342979
5	60046272	378473472	278215560	64172304	6410250	310716	7203	64			787635841
6	560431872	5099687424	5106258144	1588833792	218098125	15268176	569037	10752	81		12589157403
7	5043886848	64704407040	86163670944	35090025984	6344772525	600677532	31959711	962304	15309	100	197980378297

All 15 rows

Technical details

Node id: n-628495dc49e8eab7 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-12a0c7a551b6036d

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:          336,           168,              9;
  1:         4536,          4452,            567,             16;
  2:        54432,         92736,          20979,           1344,            25;
  3:       598752,       1649088,         586845,          63840,          2625,           36;
  4:      6158592,      26127360,       13662432,        2237760,        152250,         4536,          49;
  5:     60046272,     378473472,      278215560,       64172304,       6410250,       310716,        7203,         64;
  6:    560431872,    5099687424,     5106258144,     1588833792,     218098125,     15268176,      569037,      10752,        81;
  7:   5043886848,   64704407040,    86163670944,    35090025984,    6344772525,    600677532,    31959711,     962304,     15309,     100;
  8:  44019376128,  780121165824,  1355659224192,   706462248960,  163353294000,  20055075768,  1426777443,   60802560,   1530900,   21000,   121;
  9: 374164697088, 9000739657728, 20095876906560, 13166442971136, 3808678887000, 588908652228, 53732961009, 3036039552, 107469180, 2320500, 27951, 144;

%F Formula

The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(36*n^2+288*n+252+(48*n+336)*x-22*x^2)*P(n-2,x) - ((432*n+3024)*x-(24*n+24)*x^2-26*x^3)*d/dx P(n-2,x) - (1296*x^2-144*x^3-5*x^4)*d^2/dx^2 P(n-2,x) + (12*n^2+78*n+108+(8*n+16)*x)*P(n-1,x) + ((72*n+144)*x-(4*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 336 + 168*x + 9*x^2, P(1,x) = 4536 + 4452*x + 567*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 336 + 168*x + 9*x^2;
P[1, x_] := 4536 + 4452*x + 567*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(36*n^2 + 288*n + 252 + (48*n + 336)*x - 22*x^2)*P[n - 2, x] - ((432*n + 3024)*x - (24*n + 24)*x^2 - 26*x^3)*D[P[n - 2, x], x] - (1296*x^2 - 144*x^3 - 5*x^4)*D[P[n - 2, x], {x, 2}] + (12*n^2 + 78*n + 108 + (8*n + 16)*x)*P[n - 1, x] + ((72*n + 144)*x - (4*n + 8)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 336 + 168*x + 9*x^2; v[2] = 4536 + 4452*x + 567*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(36*n^2 + 288*n + 252 + (48*n + 336)*x - 22*x^2)*v[i - 2] - ((432*n + 3024)*x - (24*n + 24)*x^2 - 26*x^3)*deriv(v[i - 2], x) - (1296*x^2 - 144*x^3 - 5*x^4)*deriv(deriv(v[i - 2], x), x) + (12*n^2 + 78*n + 108 + (8*n + 16)*x)*v[i - 1] + ((72*n + 144)*x - (4*n + 8)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n + 6)); 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 A convolution triangle of numbers obtained from A036224.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.