one_plus_derivative of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.

nonnegative coefficients

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

Recurrence

\(\left(-12 + 12\,n\right)\,P_{n}(x) = \left(-36\,x + 25\,x^{2} - 55\,x^{3} + 58\,x^{4} - 8\,x^{5} - 48\,n + 12\,n x - 52\,n x^{2} + 64\,n x^{3} - 178\,n x^{4} + 16\,n x^{5} + 48\,n^{2} x^{2} + 168\,n^{2} x^{4} - 8\,n^{2} x^{5} - 48\,n^{3} x^{4}\right)\,P_{n-2}(x) + \left(9\,x^{2} + 2\,x^{3} - 58\,x^{4} + 47\,x^{5} - 4\,x^{6} + 12\,n x + 24\,n x^{2} - 32\,n x^{3} + 10\,n x^{4} - 96\,n x^{5} + 4\,n x^{6} + 40\,n^{2} x^{5}\right)\,P_{n-2}'(x) + \left(9\,x^{3} - 8\,x^{4} - 9\,x^{5} + 8\,x^{6} + 12\,n x^{2} - 4\,n x^{4} - 8\,n x^{6}\right)\,P_{n-2}''(x) + \left(-24 + 6\,x + 19\,x^{2} - 4\,x^{3} + 48\,n - 76\,n x^{2} + 4\,n x^{3} + 48\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-12\,x - 9\,x^{2} + 20\,x^{3} - 20\,n x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2 + x\)

\(P_{1}(x) = 4 + 9\,x + 6\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	x²⁹	P_n(1)
0	2	1																													3
1	4	9	6	1																											20
2	8	39	64	44	12	1																									168
3	16	137	406	575	420	151	22	1																							1728
4	32	443	2084	4850	6336	4812	2068	454	40	1																					21120
5	64	1381	9626	33205	66424	82418	64652	31466	8920	1289	74	1																			299520
6	128	4231	41976	202704	572844	1032843	1236992	995416	531240	180429	35880	3576	140	1																	4838400
7	256	12849	177102	1155643	4396332	10790073	18023506	21037563	17286120	9904267	3845778	957897	138796	9867	270	1															87736320
8	512	38835	732748	6311774	31322568	100443742	221747732	349305774	399846400	334271180	202647348	87276970	25701032	4848546	525292	27338	528	1													1765048320
9	1024	117053	2995906	33526673	212393424	865548844	2438118728	4954717116	7450172368	8399008574	7124365116	4522260502	2115511664	709368012	162903976	23805596	1965552	76429	1042	1											39016857600
10	2048	352223	12159008	174825788	1392171732	7069020405	24790076352	63037696368	119834776368	173502689382	193219559424	165938308296	109430868792	54771044442	20377374528	5451472560	995925168	114611307	7321312	216028	2068	1									940120473600
11	4096	1058761	49117238	900337975	8910848740	55563653723	238275613278	741765576885	1735756457904	3123075157290	4382092899276	4831375348566	4193723113944	2856209045574	1512404062860	612750919002	185263933872	40254276765	5937634046	544968035	27270628	617423	4118	1							24524881920000
12	8192	3180427	197810292	4597306602	56077678416	424720794336	2197338303220	8239003681190	23272752044760	50826621298035	87323597744616	119328208975716	130467463397376	114304469419896	80025406011528	44445556178508	19337420270160	6464853463425	1614205583140	288734124050	34781408112	2571951912	101888772	1782702	8216	1					688658684313600
13	16384	9549525	795035946	23336815405	348623370600	3180180276054	19643726953220	87611988618606	294338986310760	766678977652775	1579510673882982	2609015005180935	3485530398519888	3784063311785700	3342403812036600	2396547509855892	1386669670488720	640973368893843	233115036448790	65253193932075	13622793222984	2027163644390	201243010116	12090503070	382587400	5192529	16410	1			20710772283801600
14	32768	28665015	3191020360	117975278696	2148128860476	23447113003183	171508367980960	901002629589720	3558789471736856	10907178812866171	26521824704582664	51975005500717520	82986532021164900	108719912396160243	117319615767379968	104376453580705584	76416352425193680	45815550631706805	22310886909903448	8717190057483528	2685495030844868	636597659342365	112209130854240	14007941052536	1154507311256	56760531033	1445436760	15233600	32796	1	664173042204672000

Technical details

Node id: n-ad4e72cd8cf291a3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f289c062e1488181

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    2,      1;
  1:    4,      9,       6,        1;
  2:    8,     39,      64,       44,        12,         1;
  3:   16,    137,     406,      575,       420,       151,         22,          1;
  4:   32,    443,    2084,     4850,      6336,      4812,       2068,        454,         40,          1;
  5:   64,   1381,    9626,    33205,     66424,     82418,      64652,      31466,       8920,       1289,         74,          1;
  6:  128,   4231,   41976,   202704,    572844,   1032843,    1236992,     995416,     531240,     180429,      35880,       3576,        140,         1;
  7:  256,  12849,  177102,  1155643,   4396332,  10790073,   18023506,   21037563,   17286120,    9904267,    3845778,     957897,     138796,      9867,       270,        1;
  8:  512,  38835,  732748,  6311774,  31322568, 100443742,  221747732,  349305774,  399846400,  334271180,  202647348,   87276970,   25701032,   4848546,    525292,    27338,     528,     1;
  9: 1024, 117053, 2995906, 33526673, 212393424, 865548844, 2438118728, 4954717116, 7450172368, 8399008574, 7124365116, 4522260502, 2115511664, 709368012, 162903976, 23805596, 1965552, 76429, 1042, 1;

%F Formula

The row polynomials satisfy (12*n-12)*P(n,x) = -(48*n-(12*n-36)*x-(48*n^2-52*n+25)*x^2-(64*n-55)*x^3+(48*n^3-168*n^2+178*n-58)*x^4+(8*n^2-16*n+8)*x^5)*P(n-2,x) + (12*n*x+(24*n+9)*x^2-(32*n-2)*x^3+(10*n-58)*x^4+(40*n^2-96*n+47)*x^5+(4*n-4)*x^6)*d/dx P(n-2,x) + (12*n*x^2+9*x^3-(4*n+8)*x^4-9*x^5-(8*n-8)*x^6)*d^2/dx^2 P(n-2,x) + (48*n-24+6*x+(48*n^2-76*n+19)*x^2+(4*n-4)*x^3)*P(n-1,x) - (12*x+9*x^2+(20*n-20)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 4 + 9*x + 6*x^2 + x^3.

%t Mathematica

P[0, x_] := 2 + x;
P[1, x_] := 4 + 9*x + 6*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(48*n - (12*n - 36)*x - (48*n^2 - 52*n + 25)*x^2 - (64*n - 55)*x^3 + (48*n^3 - 168*n^2 + 178*n - 58)*x^4 + (8*n^2 - 16*n + 8)*x^5)*P[n - 2, x] + (12*n*x + (24*n + 9)*x^2 - (32*n - 2)*x^3 + (10*n - 58)*x^4 + (40*n^2 - 96*n + 47)*x^5 + (4*n - 4)*x^6)*D[P[n - 2, x], x] + (12*n*x^2 + 9*x^3 - (4*n + 8)*x^4 - 9*x^5 - (8*n - 8)*x^6)*D[P[n - 2, x], {x, 2}] + (48*n - 24 + 6*x + (48*n^2 - 76*n + 19)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] - (12*x + 9*x^2 + (20*n - 20)*x^3)*D[P[n - 1, x], x])/(12*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; v[2] = 4 + 9*x + 6*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n - (12*n - 36)*x - (48*n^2 - 52*n + 25)*x^2 - (64*n - 55)*x^3 + (48*n^3 - 168*n^2 + 178*n - 58)*x^4 + (8*n^2 - 16*n + 8)*x^5)*v[i - 2] + (12*n*x + (24*n + 9)*x^2 - (32*n - 2)*x^3 + (10*n - 58)*x^4 + (40*n^2 - 96*n + 47)*x^5 + (4*n - 4)*x^6)*deriv(v[i - 2], x) + (12*n*x^2 + 9*x^3 - (4*n + 8)*x^4 - 9*x^5 - (8*n - 8)*x^6)*deriv(deriv(v[i - 2], x), x) + (48*n - 24 + 6*x + (48*n^2 - 76*n + 19)*x^2 + (4*n - 4)*x^3)*v[i - 1] - (12*x + 9*x^2 + (20*n - 20)*x^3)*deriv(v[i - 1], x))/(12*n - 12)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
    
    Product of real-rooted factors (elementary) polytool ? step ›
    
    from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
  - ↑ Product of real-rooted factors (elementary) (above)

References

[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.