A142460 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 - 40\,n + 25\,n^{2}\right)\,P_{n}(x) = \left(-16 - 3744\,x + 144\,x^{2} - 60\,n + 1540\,n x - 6920\,n x^{2} + 100\,n^{2} + 1200\,n^{2} x + 20225\,n^{2} x^{2} - 500\,n^{3} x - 19250\,n^{3} x^{2} + 5625\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(640\,x - 3336\,x^{2} + 304\,x^{3} + 150\,n x - 715\,n x^{2} - 8920\,n x^{3} - 250\,n^{2} x + 1800\,n^{2} x^{2} + 22475\,n^{2} x^{3} + 500\,n^{3} x^{2} - 16750\,n^{3} x^{3} + 2500\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(600\,x^{2} - 680\,x^{3} + 80\,x^{4} + 375\,n x^{2} + 1425\,n x^{3} - 1800\,n x^{4} - 625\,n^{2} x^{2} - 3500\,n^{2} x^{3} + 4125\,n^{2} x^{4} + 2500\,n^{3} x^{3} - 2500\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(32 - 256\,x + 20\,n + 960\,n x - 75\,n^{2} - 1200\,n^{2} x + 500\,n^{3} x\right)\,P_{n-1}(x) + \left(-40\,x - 64\,x^{2} - 275\,n x - 160\,n x^{2} + 250\,n^{2} x + 700\,n^{2} x^{2} - 500\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	48	9													58
3	1	332	747	16												1096
4	1	2056	16434	8224	25											26740
5	1	12420	254790	452960	77625	36										797832
6	1	74624	3389535	14494720	9415375	671616	49									28045920
7	1	447868	41650389	358458160	560090875	166601556	5486383	64								1132735296
8	1	2687352	489393072	7660504096	23523816750	17236134216	2664473392	42997632	81							51620006592
9	1	16124276	5600999304	149539061504	805208130650	1159499708136	457963375856	39829328384	326516589	100						2617983244800
10	1	96745840	63098456805	2751088939520	24152669035250	60293984823072	47339231309090	11004355758080	567886111245	2418646000	121					146174829825024
11	1	580475244	704096219295	48616599439440	661954802588250	2645818751133432	3601253300153838	1694604294625920	246121534662165	7823291325500	17559376131	144				8906914809999360
12	1	3482851688	7811667946494	835445015186720	17015551755942375	103107024495644112	223277704609537956	183301376881145088	55130387689253295	5221531344917000	105023535725086	125382660768	169			588001985860810752
13	1	20897110372	86367186724122	14075378136802976	417374197341335375	3685380812557584732	11973853043182238124	15639318260482923264	8292106828254565647	1669496789365341500	106445047159572506	1381874987585952	882902913217	196		41799519502454697984
14	1	125382662496	952907182363899	233804383444937984	9886489839444530025	123466767958387258272	576854168017210152027	1126030914754770475008	953575257334571883963	342963244328853495200	47850610822911525321	2104239451004441856	17893479313277659	6143750462304	225	3182984349546227466240

Technical details

Node id: n-afeb8e974eb4c3ed · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f9758221370c5dd6

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,       48,          9;
  3: 1,      332,        747,           16;
  4: 1,     2056,      16434,         8224,           25;
  5: 1,    12420,     254790,       452960,        77625,            36;
  6: 1,    74624,    3389535,     14494720,      9415375,        671616,           49;
  7: 1,   447868,   41650389,    358458160,    560090875,     166601556,      5486383,          64;
  8: 1,  2687352,  489393072,   7660504096,  23523816750,   17236134216,   2664473392,    42997632,        81;
  9: 1, 16124276, 5600999304, 149539061504, 805208130650, 1159499708136, 457963375856, 39829328384, 326516589, 100;

%F Formula

The row polynomials satisfy (25*n^2-40*n+16)*P(n,x) = (100*n^2-60*n-16-(500*n^3-1200*n^2-1540*n+3744)*x+(5625*n^4-19250*n^3+20225*n^2-6920*n+144)*x^2)*P(n-2,x) - ((250*n^2-150*n-640)*x-(500*n^3+1800*n^2-715*n-3336)*x^2-(2500*n^4-16750*n^3+22475*n^2-8920*n+304)*x^3)*d/dx P(n-2,x) - ((625*n^2-375*n-600)*x^2-(2500*n^3-3500*n^2+1425*n-680)*x^3+(2500*n^3-4125*n^2+1800*n-80)*x^4)*d^2/dx^2 P(n-2,x) - (75*n^2-20*n-32-(500*n^3-1200*n^2+960*n-256)*x)*P(n-1,x) + ((250*n^2-275*n-40)*x-(500*n^3-700*n^2+160*n+64)*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[((100*n^2 - 60*n - 16 - (500*n^3 - 1200*n^2 - 1540*n + 3744)*x + (5625*n^4 - 19250*n^3 + 20225*n^2 - 6920*n + 144)*x^2)*P[n - 2, x] - ((250*n^2 - 150*n - 640)*x - (500*n^3 + 1800*n^2 - 715*n - 3336)*x^2 - (2500*n^4 - 16750*n^3 + 22475*n^2 - 8920*n + 304)*x^3)*D[P[n - 2, x], x] - ((625*n^2 - 375*n - 600)*x^2 - (2500*n^3 - 3500*n^2 + 1425*n - 680)*x^3 + (2500*n^3 - 4125*n^2 + 1800*n - 80)*x^4)*D[P[n - 2, x], {x, 2}] - (75*n^2 - 20*n - 32 - (500*n^3 - 1200*n^2 + 960*n - 256)*x)*P[n - 1, x] + ((250*n^2 - 275*n - 40)*x - (500*n^3 - 700*n^2 + 160*n + 64)*x^2)*D[P[n - 1, x], x])/(25*n^2 - 40*n + 16)]];
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] = ((100*n^2 - 60*n - 16 - (500*n^3 - 1200*n^2 - 1540*n + 3744)*x + (5625*n^4 - 19250*n^3 + 20225*n^2 - 6920*n + 144)*x^2)*v[i - 2] - ((250*n^2 - 150*n - 640)*x - (500*n^3 + 1800*n^2 - 715*n - 3336)*x^2 - (2500*n^4 - 16750*n^3 + 22475*n^2 - 8920*n + 304)*x^3)*deriv(v[i - 2], x) - ((625*n^2 - 375*n - 600)*x^2 - (2500*n^3 - 3500*n^2 + 1425*n - 680)*x^3 + (2500*n^3 - 4125*n^2 + 1800*n - 80)*x^4)*deriv(deriv(v[i - 2], x), x) - (75*n^2 - 20*n - 32 - (500*n^3 - 1200*n^2 + 960*n - 256)*x)*v[i - 1] + ((250*n^2 - 275*n - 40)*x - (500*n^3 - 700*n^2 + 160*n + 64)*x^2)*deriv(v[i - 1], x))/(25*n^2 - 40*n + 16)); 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) + (m*…?
  - 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) + (m*…?
    
    ↑ 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.