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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	44019376128	390060582912	451886408064	176615562240	32670658800	3342512628	203825349	7600320	170100	2100	11							1098806698652
9	374164697088	4500369828864	6698625635520	3291610742784	761735777400	98151442038	7676137287	379504944	11941020	232050	2541	12						15732725941548
10	3108445175808	49965378011136	94308021432576	57439532602368	16348460234400	2596316377248	252972199305	15907969728	659317806	17917200	307461	3024	13					224035711548073
11	25311625003008	536231463026688	1269009727679232	946753739430912	326921415660000	62930423471328	7477434013203	581086114272	30436710822	1083820500	25895331	397656	3549	14				3175248461226515
12	202493000024064	5583556982145024	16404193953202176	14841860298080256	6148297640378880	1415530748943360	201756641341248	18968609699136	1220111661492	54626798520	1702932462	36278928	503958	4116	15			44817934351993635
13	1594632375189504	56584137694224384	204572275572105216	222530271510491136	109550124360331200	29838759511712256	5035058695109016	563422913479944	43573962566340	2383678473300	93038089914	2576908404	49508550	627690	4725	16		630314735938821595
14	12381851383824384	559559627184144384	2469845460960792576	3205783370523672576	1860391089434822400	594026697826388736	117404919328054176	15434660066563584	1412101251579249	92418383732400	4389249986154	151680047904	3777310719	66061800	770175	5376	17	8836336741117756610

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.