A371259 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-144 - 18\,x - x^{2} - 168\,n - 12\,n x - 73\,n^{2} - 2\,n^{2} x - 14\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-81\,x - 8\,x^{2} - 72\,n x - 2\,n x^{2} - 21\,n^{2} x - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-9\,x^{2} - x^{3} - 6\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(34 + 2\,x + 16\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(11\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 36 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	36	2														38
2	1764	200	3													1967
3	112896	19728	600	4												133228
4	9144576	2198592	102192	1376	5											11446741
5	914457600	284318784	18013536	368800	2700	6										1217161426
6	110649369600	42770820096	3470765760	96302848	1068500	4776	7									156988331587
7	15933509222400	7449567335424	744426319104	26295828480	393914560	2662464	7840	8								24154195290280
8	2692763058585600	1491982216187904	178670272801536	7724300516352	146711093440	1335335808	5936448	12160	9							4371287900469257
9	527781559482777600	341081524759449600	47977563161475072	2470549879627776	57189769920000	658109538816	3932470976	12147072	18036	10						919369049107424958
10	118750850883624960000	88372433457384652800	14377138040805470208	864428245214699520	23733694290654720	331871539430400	2502014162368	10373435392	23206500	25800	11					222388918705270697719
11	30400217826207989760000	25777134970951414579200	4790751375407049326592	331228113910247153664	10574013022780262400	174835782037592064	1598158317254912	8346992079360	25037058816	41910000	35816	12				61310082741811907012836
12	8785662951774109040640000	8412592166240674303180800	1768126948978371928768512	138878914064641593901056	5077174885433904660480	97309334133530873856	1049400828871705856	6634334086002688	25038527849280	56150114240	72210864	48480	13			19110436521337622009956125
13	2846554796374811329167360000	3054517097916431641529548800	719869667840709782962176000	63603103701144772148723712	2630951287130811848048640	57569247619158517862400	717982433919221925888	5345774036747915264	24389670672866304	68754019780160	118378443040	119547168	64220	14		6687233909720513995334261610
14	1027606281491306889829416960000	1227499948759322279270154240000	322044299164627661748613939200	31743728415141682763931844608	1471277110518630432803635200	36317028173192280506400768	515903143421865307852800	4434294601892130062336	23794065711860940288	81002785588608000	175185692354816	236732570880	191224852	83496	15	2610402372330403604002930717259

Technical details

Node id: n-e0b9881fafc1620d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-549c91eb45245c99

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:                 36,                  2;
  2:               1764,                200,                 3;
  3:             112896,              19728,               600,                4;
  4:            9144576,            2198592,            102192,             1376,              5;
  5:          914457600,          284318784,          18013536,           368800,           2700,            6;
  6:       110649369600,        42770820096,        3470765760,         96302848,        1068500,         4776,          7;
  7:     15933509222400,      7449567335424,      744426319104,      26295828480,      393914560,      2662464,       7840,        8;
  8:   2692763058585600,   1491982216187904,   178670272801536,    7724300516352,   146711093440,   1335335808,    5936448,    12160,     9;
  9: 527781559482777600, 341081524759449600, 47977563161475072, 2470549879627776, 57189769920000, 658109538816, 3932470976, 12147072, 18036, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4+14*n^3+73*n^2+168*n+144+(2*n^2+12*n+18)*x+x^2)*P(n-2,x) - ((2*n^3+21*n^2+72*n+81)*x+(2*n+8)*x^2)*d/dx P(n-2,x) - ((n^2+6*n+9)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+16*n+34+2*x)*P(n-1,x) + (2*n+11)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 36 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 36 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (2*n^2 + 12*n + 18)*x + x^2)*P[n - 2, x] - ((2*n^3 + 21*n^2 + 72*n + 81)*x + (2*n + 8)*x^2)*D[P[n - 2, x], x] - ((n^2 + 6*n + 9)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 16*n + 34 + 2*x)*P[n - 1, x] + (2*n + 11)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 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] = 36 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (2*n^2 + 12*n + 18)*x + x^2)*v[i - 2] - ((2*n^3 + 21*n^2 + 72*n + 81)*x + (2*n + 8)*x^2)*deriv(v[i - 2], x) - ((n^2 + 6*n + 9)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 16*n + 34 + 2*x)*v[i - 1] + (2*n + 11)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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 read by rows, (2, 3)-Lah numbers.?
  - 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.