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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	44019376128	780121165824	1355659224192	706462248960	163353294000	20055075768	1426777443	60802560	1530900	21000	121							3071159516896
9	374164697088	9000739657728	20095876906560	13166442971136	3808678887000	588908652228	53732961009	3036039552	107469180	2320500	27951	144						47091690590076
10	3108445175808	99930756022272	282924064297728	229758130409472	81742301172000	15577898263488	1770805395135	127263757824	5933860254	179172000	3382071	36288	169					714945780944509
11	25311625003008	1072462926053376	3807029183037696	3787014957723648	1634607078300000	377582540827968	52342038092421	4648688914176	273930397398	10838205000	284848641	4771872	46137	196				10761284096221537
12	202493000024064	11167113964290048	49212581859606528	59367441192321024	30741488201894400	8493184493660160	1412296489388736	151748877593088	10981004953428	546267985200	18732257082	435347136	6551454	57624	225			160759894525930197
13	1594632375189504	113168275388448768	613716826716315648	890121086041964544	547750621801656000	179032557070273536	35245410865763112	4507383307839552	392165663097060	23836784733000	1023418989054	30922900848	643611150	8787660	70875	256		2385553851009640567
14	12381851383824384	1119119254368288768	7409536382882377728	12823133482094690304	9301955447174112000	3564160186958332416	821834435296379232	123477280532508672	12708911264213241	924183837324000	48281749847694	1820160574848	49105039347	924865200	11552625	86016	289	35189281567744016764

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.