A132166 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 5\,n + n^{2}\right)\,P_{n}(x) = \left(-252 - 24\,x - 2\,x^{2} - 288\,n - 24\,n x - 36\,n^{2}\right)\,P_{n-2}(x) + \left(-216\,x - 42\,x^{2} - x^{3} - 216\,n x - 6\,n x^{2}\right)\,P_{n-2}'(x) + \left(108 + 8\,x + 78\,n + 4\,n x + 12\,n^{2}\right)\,P_{n-1}(x) + \left(72\,x + 2\,x^{2} + 36\,n x + n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 336 + 84\,x + 3\,x^{2}\)

\(P_{1}(x) = 4536 + 2226\,x + 189\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	336	84	3								423
1	4536	2226	189	4							6955
2	54432	46368	6993	336	5						108134
3	598752	824544	195615	15960	525	6					1635402
4	6158592	13063680	4554144	559440	30450	756	7				24367069
5	60046272	189236736	92738520	16043076	1282050	51786	1029	8			359399477
6	560431872	2549843712	1702086048	397208448	43619625	2544696	81291	1344	9		5255817045
7	5043886848	32352203520	28721223648	8772506496	1268954505	100112922	4565673	120288	1701	10	76263575611

All 15 rows

Technical details

Node id: n-62e7ff3e821df826 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4aebd2e3981727e0

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,            84,             3;
  1:         4536,          2226,           189,             4;
  2:        54432,         46368,          6993,           336,            5;
  3:       598752,        824544,        195615,         15960,          525,           6;
  4:      6158592,      13063680,       4554144,        559440,        30450,         756,          7;
  5:     60046272,     189236736,      92738520,      16043076,      1282050,       51786,       1029,         8;
  6:    560431872,    2549843712,    1702086048,     397208448,     43619625,     2544696,      81291,      1344,        9;
  7:   5043886848,   32352203520,   28721223648,    8772506496,   1268954505,   100112922,    4565673,    120288,     1701,     10;
  8:  44019376128,  390060582912,  451886408064,  176615562240,  32670658800,  3342512628,  203825349,   7600320,   170100,   2100,   11;
  9: 374164697088, 4500369828864, 6698625635520, 3291610742784, 761735777400, 98151442038, 7676137287, 379504944, 11941020, 232050, 2541, 12;

%F Formula

The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(36*n^2+288*n+252+(24*n+24)*x+2*x^2)*P(n-2,x) - ((216*n+216)*x+(6*n+42)*x^2+x^3)*d/dx P(n-2,x) + (12*n^2+78*n+108+(4*n+8)*x)*P(n-1,x) + ((36*n+72)*x+(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 336 + 84*x + 3*x^2, P(1,x) = 4536 + 2226*x + 189*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 336 + 84*x + 3*x^2;
P[1, x_] := 4536 + 2226*x + 189*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(36*n^2 + 288*n + 252 + (24*n + 24)*x + 2*x^2)*P[n - 2, x] - ((216*n + 216)*x + (6*n + 42)*x^2 + x^3)*D[P[n - 2, x], x] + (12*n^2 + 78*n + 108 + (4*n + 8)*x)*P[n - 1, x] + ((36*n + 72)*x + (n + 2)*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 + 84*x + 3*x^2; v[2] = 4536 + 2226*x + 189*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(36*n^2 + 288*n + 252 + (24*n + 24)*x + 2*x^2)*v[i - 2] - ((216*n + 216)*x + (6*n + 42)*x^2 + x^3)*deriv(v[i - 2], x) + (12*n^2 + 78*n + 108 + (4*n + 8)*x)*v[i - 1] + ((36*n + 72)*x + (n + 2)*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.