theta_plus_one 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

\(\left(1 - 2\,x + 4\,n x\right)\,P_{n}(x) = \left(-2\,x^{2} + 6\,x^{4} - 4\,x^{5} + 2\,n x^{2} - 8\,n x^{3} + 14\,n x^{4} + 4\,n x^{5} + 8\,n^{2} x^{3} - 20\,n^{2} x^{4} + 16\,n^{2} x^{5} - 16\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-x^{2} - x^{3} + 3\,x^{4} + x^{5} - 2\,x^{6} - 6\,n x^{3} + 6\,n x^{5} - 8\,n^{2} x^{4} + 8\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(1 - 3\,x^{2} + 2\,x^{3} + 4\,n x + 12\,n x^{2} - 12\,n x^{3} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x - 2\,x^{2} - x^{3} + 2\,x^{4} + 4\,n x^{2} - 4\,n x^{4}\right)\,P_{n-1}'(x)\)

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

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

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	1	2																													3
1	1	6	9	4																											20
2	1	14	48	64	35	6																									168
3	1	30	183	460	575	366	105	8																							1728
4	1	62	618	2504	5280	6336	4382	1648	279	10																					21120
5	1	126	1977	11956	37770	70644	82418	60432	26901	6590	693	12																			299520
6	1	254	6156	53232	236745	630594	1082368	1236992	945891	473490	146388	24624	1651	14																	4838400
7	1	510	18891	227740	1373385	4945986	11626699	18700056	21037563	16609570	9067641	3296124	740155	88158	3825	16															87736320
8	1	1022	57486	951448	7592390	35764908	110230078	235429232	359861760	399846400	323715194	188965848	77490634	21258692	3567930	306592	8687	18													1765048320
9	1	2046	174045	3917188	40684220	245107896	962147116	2629336160	5204306358	7635462340	8399008574	6939075144	4272671260	1924294232	612769740	130189504	16648049	1044270	19437	20											39016857600
10	1	4094	525264	15978560	213538935	1619356734	7932315552	27036443904	67115408490	124863898620	177117929472	193219559424	162323068206	104401746540	50693332320	18131006976	4588177413	768740166	75898160	3501760	42987	22									940120473600
11	1	8190	1581999	64786540	1105176675	10427776086	62796639549	262062474024	797633738586	1830144985500	3234066724614	4459731090984	4831375348566	4116084922236	2745217478250	1418015535264	556882757301	161477073126	33021290939	4420706700	340129335	11601326	94185	24							24524881920000
12	1	16382	4758354	261632232	5664873180	65933644536	482687162510	2432437707760	8926817588685	24756537137550	53186064342708	89987455621296	121148358868992	130467463397376	112484319526620	77361548134848	42086113133835	17853635177370	5777039555930	1379106178600	230767755876	24925441992	1504385334	38066832	204775	26					688658684313600
13	1	32766	14299713	1053692500	28841740350	411426027036	3630210816806	21857286180528	95489806326795	315254453144450	808668885572163	1642903986162708	2678115201306324	3531792424333320	3784063311785700	3296141786223168	2327447313730503	1323276358208994	598983460974455	212199569615100	57375376223886	11409233995676	1577133103638	138440353680	6585578125	123930846	442341	28			20710772283801600
14	1	65534	42948372	4235605648	146145471605	2542679719386	26860555224944	191624166635392	986680809855333	3832398078490910	11576332905518788	27784833604451952	53797902925375401	84913480395838794	109987139781918720	117319615767379968	103109226194947107	74489404050519786	43992653207048924	21047878010034160	8048035964830911	2411886424090814	550919479076752	92093332199808	10594498830775	759956452346	28590338124	400851472	950243	30	664173042204672000

Technical details

Node id: n-dbfee4700b3112f1 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-d4a70096ed1391b3

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: 1,    2;
  1: 1,    6,      9,       4;
  2: 1,   14,     48,      64,       35,         6;
  3: 1,   30,    183,     460,      575,       366,       105,          8;
  4: 1,   62,    618,    2504,     5280,      6336,      4382,       1648,        279,         10;
  5: 1,  126,   1977,   11956,    37770,     70644,     82418,      60432,      26901,       6590,        693,         12;
  6: 1,  254,   6156,   53232,   236745,    630594,   1082368,    1236992,     945891,     473490,     146388,      24624,       1651,         14;
  7: 1,  510,  18891,  227740,  1373385,   4945986,  11626699,   18700056,   21037563,   16609570,    9067641,    3296124,     740155,      88158,      3825,        16;
  8: 1, 1022,  57486,  951448,  7592390,  35764908, 110230078,  235429232,  359861760,  399846400,  323715194,  188965848,   77490634,   21258692,   3567930,    306592,     8687,      18;
  9: 1, 2046, 174045, 3917188, 40684220, 245107896, 962147116, 2629336160, 5204306358, 7635462340, 8399008574, 6939075144, 4272671260, 1924294232, 612769740, 130189504, 16648049, 1044270, 19437, 20;

%F Formula

The row polynomials satisfy (1+(4*n-2)*x)*P(n,x) = ((2*n-2)*x^2+(8*n^2-8*n)*x^3-(20*n^2-14*n-6)*x^4-(16*n^3-16*n^2-4*n+4)*x^5)*P(n-2,x) - (x^2+(6*n+1)*x^3+(8*n^2-3)*x^4-(6*n+1)*x^5-(8*n^2-2)*x^6)*d/dx P(n-2,x) + (1+4*n*x+(12*n-3)*x^2+(16*n^2-12*n+2)*x^3)*P(n-1,x) + (x+(4*n-2)*x^2-x^3-(4*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 2*x, P(1,x) = 1 + 6*x + 9*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 1 + 2*x;
P[1, x_] := 1 + 6*x + 9*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 2)*x^2 + (8*n^2 - 8*n)*x^3 - (20*n^2 - 14*n - 6)*x^4 - (16*n^3 - 16*n^2 - 4*n + 4)*x^5)*P[n - 2, x] - (x^2 + (6*n + 1)*x^3 + (8*n^2 - 3)*x^4 - (6*n + 1)*x^5 - (8*n^2 - 2)*x^6)*D[P[n - 2, x], x] + (1 + 4*n*x + (12*n - 3)*x^2 + (16*n^2 - 12*n + 2)*x^3)*P[n - 1, x] + (x + (4*n - 2)*x^2 - x^3 - (4*n - 2)*x^4)*D[P[n - 1, x], x])/(1 + (4*n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; v[2] = 1 + 6*x + 9*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 2)*x^2 + (8*n^2 - 8*n)*x^3 - (20*n^2 - 14*n - 6)*x^4 - (16*n^3 - 16*n^2 - 4*n + 4)*x^5)*v[i - 2] - (x^2 + (6*n + 1)*x^3 + (8*n^2 - 3)*x^4 - (6*n + 1)*x^5 - (8*n^2 - 2)*x^6)*deriv(v[i - 2], x) + (1 + 4*n*x + (12*n - 3)*x^2 + (16*n^2 - 12*n + 2)*x^3)*v[i - 1] + (x + (4*n - 2)*x^2 - x^3 - (4*n - 2)*x^4)*deriv(v[i - 1], x))/(1 + (4*n - 2)*x)); 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

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), 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

[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.