A257618 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 24\,n + 16\,n^{2}\right)\,P_{n}(x) = \left(-36 - 5712\,x - 252\,x^{2} - 96\,n + 2368\,n x - 8256\,n x^{2} + 192\,n^{2} + 2304\,n^{2} x + 28544\,n^{2} x^{2} - 1024\,n^{3} x - 29696\,n^{3} x^{2} + 9216\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(1056\,x - 5232\,x^{2} - 144\,x^{3} + 256\,n x - 832\,n x^{2} - 11136\,n x^{3} - 512\,n^{2} x + 2816\,n^{2} x^{2} + 32768\,n^{2} x^{3} + 1024\,n^{3} x^{2} - 26624\,n^{3} x^{3} + 4096\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(960\,x^{2} - 960\,x^{3} + 512\,n x^{2} + 1792\,n x^{3} - 2304\,n x^{4} - 1024\,n^{2} x^{2} - 5120\,n^{2} x^{3} + 6144\,n^{2} x^{4} + 4096\,n^{3} x^{3} - 4096\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(36 - 216\,x + 864\,n x - 64\,n^{2} - 1152\,n^{2} x + 512\,n^{3} x\right)\,P_{n-1}(x) + \left(-48\,x - 72\,x^{2} - 256\,n x - 96\,n x^{2} + 256\,n^{2} x + 640\,n^{2} x^{2} - 512\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	4	160	36													200
3	8	1888	4248	128												6272
4	16	19712	152928	78848	400											251904
5	32	199296	3905856	6943744	1245600	1152										12295680
6	64	1998336	85551552	361074688	237643200	17985024	3136									704256000
7	128	19996160	1728770688	14559057920	22748528000	6915082752	244952960	8192								46216396800
8	256	199991296	33367440384	507617050624	1546899904000	1142138363904	181667175424	3199860736	20736							3415089807360
9	512	1999980544	626712791040	16164038017024	85906965683200	123706030583808	49502366427136	4456624291840	40499606016	51200						280405237432320
10	1024	19999956992	11577827349504	484886040674304	4183652303308800	10391306569039872	8199958514485248	1939544162697216	104200446145536	499998924800	123904					25315645862707200
11	2048	199999905792	211730885130240	13965502133207040	186186975748608000	737657841737908224	1004034284587708416	476638657916436480	70700354549360640	2352565390336000	6049997150208	294912				2491754162946048000
12	4096	1999999795200	3848055914962944	390957429684961280	7770549582939648000	46531965567515885568	100403428458770841600	82723494342250463232	25176580648724459520	2443483935531008000	51734973967835136	71999992627200	692224			265496116995293184000
13	8192	19999999557632	69670006427860992	10725854211856924672	309403138627271680000	2692855586198251339776	8693611593574101712896	11354921265076377747456	6058925068946065514496	1237612554509086720000	81114272477167992832	1114720102845775872	844999981309952	1605632		30440354588729435750400
14	16384	199999999049728	1258470115603955712	290019410536737800192	11893956784221405593600	146069933072448626491392	676588773172181331001344	1317170866748859818704896	1118442665856054853287936	405749814090135073587200	57566750835631603073024	2610174694830640201728	23631272170785390592	9799999953436672	3686400	3736407854407186435276800

Technical details

Node id: n-f1477600e3507b5d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-fffff3ff3ecdc6fc

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:   2,          8;
  2:   4,        160,           36;
  3:   8,       1888,         4248,            128;
  4:  16,      19712,       152928,          78848,            400;
  5:  32,     199296,      3905856,        6943744,        1245600,            1152;
  6:  64,    1998336,     85551552,      361074688,      237643200,        17985024,           3136;
  7: 128,   19996160,   1728770688,    14559057920,    22748528000,      6915082752,      244952960,          8192;
  8: 256,  199991296,  33367440384,   507617050624,  1546899904000,   1142138363904,   181667175424,    3199860736,       20736;
  9: 512, 1999980544, 626712791040, 16164038017024, 85906965683200, 123706030583808, 49502366427136, 4456624291840, 40499606016, 51200;

%F Formula

The row polynomials satisfy (16*n^2-24*n+9)*P(n,x) = (192*n^2-96*n-36-(1024*n^3-2304*n^2-2368*n+5712)*x+(9216*n^4-29696*n^3+28544*n^2-8256*n-252)*x^2)*P(n-2,x) - ((512*n^2-256*n-1056)*x-(1024*n^3+2816*n^2-832*n-5232)*x^2-(4096*n^4-26624*n^3+32768*n^2-11136*n-144)*x^3)*d/dx P(n-2,x) - ((1024*n^2-512*n-960)*x^2-(4096*n^3-5120*n^2+1792*n-960)*x^3+(4096*n^3-6144*n^2+2304*n)*x^4)*d^2/dx^2 P(n-2,x) - (64*n^2-36-(512*n^3-1152*n^2+864*n-216)*x)*P(n-1,x) + ((256*n^2-256*n-48)*x-(512*n^3-640*n^2+96*n+72)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((192*n^2 - 96*n - 36 - (1024*n^3 - 2304*n^2 - 2368*n + 5712)*x + (9216*n^4 - 29696*n^3 + 28544*n^2 - 8256*n - 252)*x^2)*P[n - 2, x] - ((512*n^2 - 256*n - 1056)*x - (1024*n^3 + 2816*n^2 - 832*n - 5232)*x^2 - (4096*n^4 - 26624*n^3 + 32768*n^2 - 11136*n - 144)*x^3)*D[P[n - 2, x], x] - ((1024*n^2 - 512*n - 960)*x^2 - (4096*n^3 - 5120*n^2 + 1792*n - 960)*x^3 + (4096*n^3 - 6144*n^2 + 2304*n)*x^4)*D[P[n - 2, x], {x, 2}] - (64*n^2 - 36 - (512*n^3 - 1152*n^2 + 864*n - 216)*x)*P[n - 1, x] + ((256*n^2 - 256*n - 48)*x - (512*n^3 - 640*n^2 + 96*n + 72)*x^2)*D[P[n - 1, x], x])/(16*n^2 - 24*n + 9)]];
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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((192*n^2 - 96*n - 36 - (1024*n^3 - 2304*n^2 - 2368*n + 5712)*x + (9216*n^4 - 29696*n^3 + 28544*n^2 - 8256*n - 252)*x^2)*v[i - 2] - ((512*n^2 - 256*n - 1056)*x - (1024*n^3 + 2816*n^2 - 832*n - 5232)*x^2 - (4096*n^4 - 26624*n^3 + 32768*n^2 - 11136*n - 144)*x^3)*deriv(v[i - 2], x) - ((1024*n^2 - 512*n - 960)*x^2 - (4096*n^3 - 5120*n^2 + 1792*n - 960)*x^3 + (4096*n^3 - 6144*n^2 + 2304*n)*x^4)*deriv(deriv(v[i - 2], x), x) - (64*n^2 - 36 - (512*n^3 - 1152*n^2 + 864*n - 216)*x)*v[i - 1] + ((256*n^2 - 256*n - 48)*x - (512*n^3 - 640*n^2 + 96*n + 72)*x^2)*deriv(v[i - 1], x))/(16*n^2 - 24*n + 9)); 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.?
  - 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142459 → (θ+1)²

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.