A013612 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 25\,x + 100\,x^{2} + 15\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 45\,x + 425\,x^{2} + 1125\,x^{3} + 15\,n x + 225\,n x^{2} + 750\,n x^{3} + 25\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 40\,x + 225\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	40	225															266
1	1	60	675	2000														2736
2	1	80	1350	8000	15625													25056
3	1	100	2250	20000	78125	112500												212976
4	1	120	3375	40000	234375	675000	765625											1718496
5	1	140	4725	70000	546875	2362500	5359375	5000000										13343616
6	1	160	6300	112000	1093750	6300000	21437500	40000000	31640625									100590336
7	1	180	8100	168000	1968750	14175000	64312500	180000000	284765625	195312500								740710656
8	1	200	10125	240000	3281250	28350000	160781250	600000000	1423828125	1953125000	1181640625							5351256576
9	1	220	12375	330000	5156250	51975000	353718750	1650000000	5220703125	10742187500	12998046875	7031250000						38053380096
10	1	240	14850	440000	7734375	89100000	707437500	3960000000	15662109375	42968750000	77988281250	84375000000	41259765625					267018633216
11	1	260	17550	572000	11171875	144787500	1313812500	8580000000	40721484375	139648437500	337949218750	548437500000	536376953125	239257812500				1852441767936
12	1	280	20475	728000	15640625	225225000	2299171875	17160000000	95016796875	391015625000	1182822265625	2559375000000	3754638671875	3349609375000	1373291015625			12725469536256
13	1	300	23625	910000	21328125	337837500	3831953125	32175000000	203607421875	977539062500	3548466796875	9597656250000	18773193359375	25122070312500	20599365234375	7812500000000		86670765490176
14	1	320	27000	1120000	28437500	491400000	6131125000	57200000000	407214843750	2234375000000	9462578125000	30712500000000	75092773437500	133984375000000	164794921875000	125000000000000	44097900390625	585850490781696

Technical details

Node id: n-a307d1b79e3601d5 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e325ebfab08c4014

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: 1,  40,   225;
  1: 1,  60,   675,   2000;
  2: 1,  80,  1350,   8000,   15625;
  3: 1, 100,  2250,  20000,   78125,   112500;
  4: 1, 120,  3375,  40000,  234375,   675000,    765625;
  5: 1, 140,  4725,  70000,  546875,  2362500,   5359375,    5000000;
  6: 1, 160,  6300, 112000, 1093750,  6300000,  21437500,   40000000,   31640625;
  7: 1, 180,  8100, 168000, 1968750, 14175000,  64312500,  180000000,  284765625,   195312500;
  8: 1, 200, 10125, 240000, 3281250, 28350000, 160781250,  600000000, 1423828125,  1953125000,  1181640625;
  9: 1, 220, 12375, 330000, 5156250, 51975000, 353718750, 1650000000, 5220703125, 10742187500, 12998046875, 7031250000;

%F Formula

The row polynomials satisfy (1+(15*n+25)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (1+(15*n+45)*x+(25*n^2+225*n+425)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 40*x + 225*x^2.

%t Mathematica

P[0, x_] := 1 + 40*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (15*n + 45)*x + (25*n^2 + 225*n + 425)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(1 + (15*n + 25)*x + (25*n^2 + 100*n + 100)*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 + 40*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (15*n + 45)*x + (25*n^2 + 225*n + 425)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(1 + (15*n + 25)*x + (25*n^2 + 100*n + 100)*x^2)); 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 of coefficients in expansion of (1+5x)^n.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - 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
[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.