A256890 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-6 - 6\,x - 6\,x^{2} - 2\,n - 8\,n x - 11\,n x^{2} - 2\,n^{2} x - 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - 3\,x^{2} + 6\,x^{3} - n x - 4\,n x^{2} + 5\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(7 + 8\,x + 3\,n + 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 4 + 24\,x + 12\,x^{2}\)

\(P_{2}(x) = 8 + 104\,x + 156\,x^{2} + 32\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	4	24	12															40
2	8	104	156	32														300
3	16	392	1248	784	80													2520
4	32	1368	7932	10576	3420	192												23520
5	64	4552	44040	105760	73400	13656	448											241920
6	128	14680	223956	881120	1101400	447912	51380	1024										2721600
7	256	46344	1071984	6495856	13216800	9743784	2501296	185376	2304									33264000
8	512	144152	4913580	43913776	136139540	163367448	76849108	13102880	648684	5120								439084800
9	1024	443720	21816600	278531840	1255975000	2287144272	1758365000	557063680	65449800	2218600	11264							6227020800
10	2048	1355736	94587780	1683547200	10669333200	28067369904	32745264888	17070933120	3787981200	315292600	7456548	24576						94443148800
11	4096	4120456	402754368	9805023440	85060339200	311700387888	523924238208	415600517184	153108610560	24512558600	1476766016	24722736	53248					1525620096000
12	8192	12476056	1691366364	55469187088	645181107500	3202626785616	7464247978488	8530569118272	4803940178424	1290362215000	152540264492	6765465456	81094364	114688				26153487360000
13	16384	37673928	7027462632	306662952416	4703124451320	30934778118312	97077962993424	153550244128896	124814523848688	51557963530520	10346873792904	919988857248	30452338072	263717496	245760			474249904128000
14	32768	113546072	28957513908	1664494064544	33202019684680	284268438927192	1173620023132396	2491414631370624	2802841460291952	1676600033046280	521158804699852	79684847243232	5409605709768	135135064904	851595540	524288		9070029416448000
15	65536	341752328	118555161360	8901620600880	228340764237600	2507830579571352	13368718330039856	37176811973142880	56056829205839040	46471014966428600	21007985947205488	5015661159142704	593685987017760	31155672103080	592775806800	2734018624	1114112	182467650613248000

Technical details

Node id: n-ca40f10a3c367093 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7c3b635378939853

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 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    4,      24,       12;
   2:    8,     104,      156,         32;
   3:   16,     392,     1248,        784,          80;
   4:   32,    1368,     7932,      10576,        3420,         192;
   5:   64,    4552,    44040,     105760,       73400,       13656,         448;
   6:  128,   14680,   223956,     881120,     1101400,      447912,       51380,        1024;
   7:  256,   46344,  1071984,    6495856,    13216800,     9743784,     2501296,      185376,       2304;
   8:  512,  144152,  4913580,   43913776,   136139540,   163367448,    76849108,    13102880,     648684,      5120;
   9: 1024,  443720, 21816600,  278531840,  1255975000,  2287144272,  1758365000,   557063680,   65449800,   2218600,   11264;
  10: 2048, 1355736, 94587780, 1683547200, 10669333200, 28067369904, 32745264888, 17070933120, 3787981200, 315292600, 7456548, 24576;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -(2*n+6+(2*n^2+8*n+6)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((n+3)*x+(n^2+4*n+3)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (3*n+7+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 4 + 24*x + 12*x^2, P(2,x) = 8 + 104*x + 156*x^2 + 32*x^3.

%t Mathematica

P[1, x_] := 4 + 24*x + 12*x^2;
P[2, x_] := 8 + 104*x + 156*x^2 + 32*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 6 + (2*n^2 + 8*n + 6)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((n + 3)*x + (n^2 + 4*n + 3)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (3*n + 7 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 4 + 24*x + 12*x^2; v[2] = 8 + 104*x + 156*x^2 + 32*x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + 6 + (2*n^2 + 8*n + 6)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((n + 3)*x + (n^2 + 4*n + 3)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (3*n + 7 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 1], x))/(n + 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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 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) = 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 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) = x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.