A142461 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 60\,n + 36\,n^{2}\right)\,P_{n}(x) = \left(-25 - 8140\,x + 825\,x^{2} - 120\,n + 3384\,n x - 17280\,n x^{2} + 180\,n^{2} + 2160\,n^{2} x + 46008\,n^{2} x^{2} - 864\,n^{3} x - 41472\,n^{3} x^{2} + 11664\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(1320\,x - 7100\,x^{2} + 1300\,x^{3} + 288\,n x - 1800\,n x^{2} - 21720\,n x^{3} - 432\,n^{2} x + 3888\,n^{2} x^{2} + 50112\,n^{2} x^{3} + 864\,n^{3} x^{2} - 35424\,n^{3} x^{3} + 5184\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(1260\,x^{2} - 1560\,x^{3} + 300\,x^{4} + 864\,n x^{2} + 3456\,n x^{3} - 4320\,n x^{4} - 1296\,n^{2} x^{2} - 7776\,n^{2} x^{3} + 9072\,n^{2} x^{4} + 5184\,n^{3} x^{3} - 5184\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(50 - 500\,x + 60\,n + 1800\,n x - 144\,n^{2} - 2160\,n^{2} x + 864\,n^{3} x\right)\,P_{n-1}(x) + \left(-60\,x - 100\,x^{2} - 504\,n x - 360\,n x^{2} + 432\,n^{2} x + 1296\,n^{2} x^{2} - 864\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	56	9													66
3	1	444	999	16												1460
4	1	3184	25974	12736	25											41920
5	1	22388	473778	842272	139925	36										1478400
6	1	156840	7418439	32006336	20606775	1411560	49									61600000
7	1	1098028	107379297	937828784	1465357475	429517188	13450843	64								2954631680
8	1	7686368	1487341692	23736539264	73267873750	53407213344	8097749212	122981888	81							160127385600
9	1	53804772	20079098100	548828277312	2981435464350	4293267068664	1680786599268	142784697600	1089546633	100						9668324556800
10	1	376633624	266960251413	11962672768128	106265021390850	266182558257168	208279441926066	47850691072512	2402642262717	9415840600	121					643219780403200
11	1	2636435612	3517091679339	250545986717520	3460367611379850	13913459645676696	18937764517726614	8858541085132416	1268389057757445	39078796437100	79752177263	144				46731746181120000
12	1	18455049552	46084042619154	5104267156279360	105691617392556375	645583211512617888	1401394574311769436	1147703487133542912	342440840351882655	31901669726746000	619574350768626	664381783872	169			3680486008815616000
13	1	129185347156	601874652768966	101978634368896096	3080938393556666875	27470680080109932348	89636149990010077188	117075787742053978368	61809030180247347783	12323753574226667500	771213422414776726	9629994444303456	5458080917341	196		312279769473351680000
14	1	904297430408	7845589179266931	2009284011649951552	86743298427202081025	1095601403253095919288	5148679296084371399499	10068517745816642139648	8511082101690491497131	3043337231258599775800	419837564387658072161	18083556104849563968	147322730144012371	44310574089992	225	28394046694568755200000

Technical details

Node id: n-67b2c6109797c507 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-bade81768928a835

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: 1,        4;
  2: 1,       56,           9;
  3: 1,      444,         999,           16;
  4: 1,     3184,       25974,        12736,            25;
  5: 1,    22388,      473778,       842272,        139925,            36;
  6: 1,   156840,     7418439,     32006336,      20606775,       1411560,            49;
  7: 1,  1098028,   107379297,    937828784,    1465357475,     429517188,      13450843,           64;
  8: 1,  7686368,  1487341692,  23736539264,   73267873750,   53407213344,    8097749212,    122981888,         81;
  9: 1, 53804772, 20079098100, 548828277312, 2981435464350, 4293267068664, 1680786599268, 142784697600, 1089546633, 100;

%F Formula

The row polynomials satisfy (36*n^2-60*n+25)*P(n,x) = (180*n^2-120*n-25-(864*n^3-2160*n^2-3384*n+8140)*x+(11664*n^4-41472*n^3+46008*n^2-17280*n+825)*x^2)*P(n-2,x) - ((432*n^2-288*n-1320)*x-(864*n^3+3888*n^2-1800*n-7100)*x^2-(5184*n^4-35424*n^3+50112*n^2-21720*n+1300)*x^3)*d/dx P(n-2,x) - ((1296*n^2-864*n-1260)*x^2-(5184*n^3-7776*n^2+3456*n-1560)*x^3+(5184*n^3-9072*n^2+4320*n-300)*x^4)*d^2/dx^2 P(n-2,x) - (144*n^2-60*n-50-(864*n^3-2160*n^2+1800*n-500)*x)*P(n-1,x) + ((432*n^2-504*n-60)*x-(864*n^3-1296*n^2+360*n+100)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((180*n^2 - 120*n - 25 - (864*n^3 - 2160*n^2 - 3384*n + 8140)*x + (11664*n^4 - 41472*n^3 + 46008*n^2 - 17280*n + 825)*x^2)*P[n - 2, x] - ((432*n^2 - 288*n - 1320)*x - (864*n^3 + 3888*n^2 - 1800*n - 7100)*x^2 - (5184*n^4 - 35424*n^3 + 50112*n^2 - 21720*n + 1300)*x^3)*D[P[n - 2, x], x] - ((1296*n^2 - 864*n - 1260)*x^2 - (5184*n^3 - 7776*n^2 + 3456*n - 1560)*x^3 + (5184*n^3 - 9072*n^2 + 4320*n - 300)*x^4)*D[P[n - 2, x], {x, 2}] - (144*n^2 - 60*n - 50 - (864*n^3 - 2160*n^2 + 1800*n - 500)*x)*P[n - 1, x] + ((432*n^2 - 504*n - 60)*x - (864*n^3 - 1296*n^2 + 360*n + 100)*x^2)*D[P[n - 1, x], x])/(36*n^2 - 60*n + 25)]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((180*n^2 - 120*n - 25 - (864*n^3 - 2160*n^2 - 3384*n + 8140)*x + (11664*n^4 - 41472*n^3 + 46008*n^2 - 17280*n + 825)*x^2)*v[i - 2] - ((432*n^2 - 288*n - 1320)*x - (864*n^3 + 3888*n^2 - 1800*n - 7100)*x^2 - (5184*n^4 - 35424*n^3 + 50112*n^2 - 21720*n + 1300)*x^3)*deriv(v[i - 2], x) - ((1296*n^2 - 864*n - 1260)*x^2 - (5184*n^3 - 7776*n^2 + 3456*n - 1560)*x^3 + (5184*n^3 - 9072*n^2 + 4320*n - 300)*x^4)*deriv(deriv(v[i - 2], x), x) - (144*n^2 - 60*n - 50 - (864*n^3 - 2160*n^2 + 1800*n - 500)*x)*v[i - 1] + ((432*n^2 - 504*n - 60)*x - (864*n^3 - 1296*n^2 + 360*n + 100)*x^2)*deriv(v[i - 1], x))/(36*n^2 - 60*n + 25)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…?
    
    ↑ Liu–Wang theorem (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.